Factoring is quick, but many quadratic equations don’t factor nicely: try finding two integers that multiply to − 4 -4 − 4 2 2 2 quadratic formula solves every quadratic equation, whether it factors or not. It comes from completing the square once, in general, so you never have to do it again.
For any quadratic equation a x 2 + b x + c = 0 ax^2 + bx + c = 0 a x 2 + b x + c = 0 a ≠ 0 a \ne 0 a = 0
x = − b ± b 2 − 4 a c 2 a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} x = 2 a − b ± b 2 − 4 a c The ± \pm ± + + + − - −
The formula comes from completing the square on a x 2 + b x + c = 0 ax^2 + bx + c = 0 a x 2 + b x + c = 0 don’t need to reproduce this general development yourself, but it’s worth following once so the formula isn’t magic. On the left is a numerical example, 2 x 2 + 8 x + 3 = 0 2x^2 + 8x + 3 = 0 2 x 2 + 8 x + 3 = 0
Step Example General case Start 2 x 2 + 8 x + 3 = 0 2x^2 + 8x + 3 = 0 2 x 2 + 8 x + 3 = 0 a x 2 + b x + c = 0 ax^2 + bx + c = 0 a x 2 + b x + c = 0 Divide by a a a x 2 + 4 x + 3 2 = 0 x^2 + 4x + \tfrac{3}{2} = 0 x 2 + 4 x + 2 3 = 0 x 2 + b a x + c a = 0 x^2 + \tfrac{b}{a}x + \tfrac{c}{a} = 0 x 2 + a b x + a c = 0 Move the constant x 2 + 4 x = − 3 2 x^2 + 4x = -\tfrac{3}{2} x 2 + 4 x = − 2 3 x 2 + b a x = − c a x^2 + \tfrac{b}{a}x = -\tfrac{c}{a} x 2 + a b x = − a c Add (half of the x x x x 2 + 4 x + 4 = 4 − 3 2 x^2 + 4x + 4 = 4 - \tfrac{3}{2} x 2 + 4 x + 4 = 4 − 2 3 x 2 + b a x + b 2 4 a 2 = b 2 4 a 2 − c a x^2 + \tfrac{b}{a}x + \tfrac{b^2}{4a^2} = \tfrac{b^2}{4a^2} - \tfrac{c}{a} x 2 + a b x + 4 a 2 b 2 = 4 a 2 b 2 − a c Write as a square ( x + 2 ) 2 = 5 2 (x + 2)^2 = \tfrac{5}{2} ( x + 2 ) 2 = 2 5 ( x + b 2 a ) 2 = b 2 − 4 a c 4 a 2 \left(x + \tfrac{b}{2a}\right)^2 = \tfrac{b^2 - 4ac}{4a^2} ( x + 2 a b ) 2 = 4 a 2 b 2 − 4 a c Square root both sides x + 2 = ± 5 2 x + 2 = \pm\sqrt{\tfrac{5}{2}} x + 2 = ± 2 5 x + b 2 a = ± b 2 − 4 a c 2 a x + \tfrac{b}{2a} = \pm\tfrac{\sqrt{b^2 - 4ac}}{2a} x + 2 a b = ± 2 a b 2 − 4 a c Solve for x x x x = − 2 ± 5 2 x = -2 \pm \sqrt{\tfrac{5}{2}} x = − 2 ± 2 5 x = − b ± b 2 − 4 a c 2 a x = \tfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} x = 2 a − b ± b 2 − 4 a c
In the “write as a square” step of the general case, the right side was put over the common denominator 4 a 2 4a^2 4 a 2 b 2 4 a 2 − 4 a c 4 a 2 = b 2 − 4 a c 4 a 2 \tfrac{b^2}{4a^2} - \tfrac{4ac}{4a^2} = \tfrac{b^2 - 4ac}{4a^2} 4 a 2 b 2 − 4 a 2 4 a c = 4 a 2 b 2 − 4 a c
Now check that the formula gives the same answer for the example, with a = 2 a = 2 a = 2 b = 8 b = 8 b = 8 c = 3 c = 3 c = 3
x = − 8 ± 8 2 − 4 ( 2 ) ( 3 ) 2 ( 2 ) = − 8 ± 40 4 x = \frac{-8 \pm \sqrt{8^2 - 4(2)(3)}}{2(2)} = \frac{-8 \pm \sqrt{40}}{4} x = 2 ( 2 ) − 8 ± 8 2 − 4 ( 2 ) ( 3 ) = 4 − 8 ± 40 Both give x ≈ − 0.42 x \approx -0.42 x ≈ − 0.42 x ≈ − 3.58 x \approx -3.58 x ≈ − 3.58
Rearrange the equation into the form a x 2 + b x + c = 0 ax^2 + bx + c = 0 a x 2 + b x + c = 0
Write down a a a b b b c c c including their signs .
Substitute, using brackets around negative numbers.
Simplify b 2 − 4 a c b^2 - 4ac b 2 − 4 a c
Write the two exact answers, then round to decimals if the question asks.
An exact answer keeps the square root, like − 1 + 5 -1 + \sqrt{5} − 1 + 5 decimal answer is rounded, like 1.24 1.24 1.24 b 2 − 4 a c b^2 - 4ac b 2 − 4 a c
Method Best when Notes Factoring the quadratic factors easily fastest; gives exact roots Quadratic formula it doesn’t factor, or you can’t see how always works; gives exact roots Graphing you want to see the roots or check them with technology (such as Desmos), read the x x x
A good habit: try factoring for a few seconds; if it doesn’t work, use the formula; then check with a graph if you can.
The expression b 2 − 4 a c b^2 - 4ac b 2 − 4 a c discriminant . If it’s negative , the formula asks for the square root of a negative number, and no real number squares to give a negative. So the equation has no real roots , and the graph of y = a x 2 + b x + c y = ax^2 + bx + c y = a x 2 + b x + c x x x zeros and the discriminant .
Solve x 2 − 3 x − 10 = 0 x^2 - 3x - 10 = 0 x 2 − 3 x − 10 = 0
Solution. a = 1 a = 1 a = 1 b = − 3 b = -3 b = − 3 c = − 10 c = -10 c = − 10
x = − ( − 3 ) ± ( − 3 ) 2 − 4 ( 1 ) ( − 10 ) 2 ( 1 ) = 3 ± 9 + 40 2 = 3 ± 49 2 = 3 ± 7 2 \begin{aligned}
x &= \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-10)}}{2(1)} \\
&= \frac{3 \pm \sqrt{9 + 40}}{2} \\
&= \frac{3 \pm \sqrt{49}}{2} \\
&= \frac{3 \pm 7}{2}
\end{aligned} x = 2 ( 1 ) − ( − 3 ) ± ( − 3 ) 2 − 4 ( 1 ) ( − 10 ) = 2 3 ± 9 + 40 = 2 3 ± 49 = 2 3 ± 7 So x = 3 + 7 2 = 5 x = \dfrac{3 + 7}{2} = 5 x = 2 3 + 7 = 5 x = 3 − 7 2 = − 2 x = \dfrac{3 - 7}{2} = -2 x = 2 3 − 7 = − 2
Check by factoring: x 2 − 3 x − 10 = ( x − 5 ) ( x + 2 ) x^2 - 3x - 10 = (x - 5)(x + 2) x 2 − 3 x − 10 = ( x − 5 ) ( x + 2 ) 49 49 49
Solve x 2 + 2 x − 4 = 0 x^2 + 2x - 4 = 0 x 2 + 2 x − 4 = 0
Solution. a = 1 a = 1 a = 1 b = 2 b = 2 b = 2 c = − 4 c = -4 c = − 4
x = − 2 ± 2 2 − 4 ( 1 ) ( − 4 ) 2 ( 1 ) = − 2 ± 4 + 16 2 = − 2 ± 20 2 \begin{aligned}
x &= \frac{-2 \pm \sqrt{2^2 - 4(1)(-4)}}{2(1)} \\
&= \frac{-2 \pm \sqrt{4 + 16}}{2} \\
&= \frac{-2 \pm \sqrt{20}}{2}
\end{aligned} x = 2 ( 1 ) − 2 ± 2 2 − 4 ( 1 ) ( − 4 ) = 2 − 2 ± 4 + 16 = 2 − 2 ± 20 The exact roots are x = − 2 + 20 2 x = \dfrac{-2 + \sqrt{20}}{2} x = 2 − 2 + 20 x = − 2 − 20 2 x = \dfrac{-2 - \sqrt{20}}{2} x = 2 − 2 − 20 simplify radicals , 20 = 2 5 \sqrt{20} = 2\sqrt{5} 20 = 2 5 − 1 ± 5 -1 \pm \sqrt{5} − 1 ± 5
With a calculator, 20 ≈ 4.472 \sqrt{20} \approx 4.472 20 ≈ 4.472
x ≈ − 2 + 4.472 2 ≈ 1.24 or x ≈ − 2 − 4.472 2 ≈ − 3.24 x \approx \frac{-2 + 4.472}{2} \approx 1.24 \qquad \text{or} \qquad x \approx \frac{-2 - 4.472}{2} \approx -3.24 x ≈ 2 − 2 + 4.472 ≈ 1.24 or x ≈ 2 − 2 − 4.472 ≈ − 3.24
The parabola y = x squared + 2x - 4 crossing the x-axis at about -3.24 and 1.24
−4
−2
2
−4
−2
2
4
≈ −3.24
≈ 1.24
y = x² + 2x − 4
The graph of y = x 2 + 2 x − 4 y = x^2 + 2x - 4 y = x 2 + 2 x − 4 x x x Solve 3 x 2 = 5 − 4 x 3x^2 = 5 - 4x 3 x 2 = 5 − 4 x
Solution. Get 0 0 0 4 x 4x 4 x 5 5 5
3 x 2 + 4 x − 5 = 0 3x^2 + 4x - 5 = 0 3 x 2 + 4 x − 5 = 0 So a = 3 a = 3 a = 3 b = 4 b = 4 b = 4 c = − 5 c = -5 c = − 5
x = − 4 ± 4 2 − 4 ( 3 ) ( − 5 ) 2 ( 3 ) = − 4 ± 16 + 60 6 = − 4 ± 76 6 \begin{aligned}
x &= \frac{-4 \pm \sqrt{4^2 - 4(3)(-5)}}{2(3)} \\
&= \frac{-4 \pm \sqrt{16 + 60}}{6} \\
&= \frac{-4 \pm \sqrt{76}}{6}
\end{aligned} x = 2 ( 3 ) − 4 ± 4 2 − 4 ( 3 ) ( − 5 ) = 6 − 4 ± 16 + 60 = 6 − 4 ± 76 With 76 ≈ 8.718 \sqrt{76} \approx 8.718 76 ≈ 8.718
x ≈ − 4 + 8.718 6 ≈ 0.79 or x ≈ − 4 − 8.718 6 ≈ − 2.12 x \approx \frac{-4 + 8.718}{6} \approx 0.79 \qquad \text{or} \qquad x \approx \frac{-4 - 8.718}{6} \approx -2.12 x ≈ 6 − 4 + 8.718 ≈ 0.79 or x ≈ 6 − 4 − 8.718 ≈ − 2.12 Check x ≈ 0.79 x \approx 0.79 x ≈ 0.79 3 ( 0.79 ) 2 ≈ 1.87 3(0.79)^2 \approx 1.87 3 ( 0.79 ) 2 ≈ 1.87 5 − 4 ( 0.79 ) = 1.84 5 - 4(0.79) = 1.84 5 − 4 ( 0.79 ) = 1.84
Solve x 2 − 4 x + 7 = 0 x^2 - 4x + 7 = 0 x 2 − 4 x + 7 = 0
Solution. a = 1 a = 1 a = 1 b = − 4 b = -4 b = − 4 c = 7 c = 7 c = 7
b 2 − 4 a c = ( − 4 ) 2 − 4 ( 1 ) ( 7 ) = 16 − 28 = − 12 b^2 - 4ac = (-4)^2 - 4(1)(7) = 16 - 28 = -12 b 2 − 4 a c = ( − 4 ) 2 − 4 ( 1 ) ( 7 ) = 16 − 28 = − 12 The formula would need − 12 \sqrt{-12} − 12 no real roots .
This makes sense graphically. Completing the square gives y = x 2 − 4 x + 7 = ( x − 2 ) 2 + 3 y = x^2 - 4x + 7 = (x - 2)^2 + 3 y = x 2 − 4 x + 7 = ( x − 2 ) 2 + 3 ( 2 , 3 ) (2, 3) ( 2 , 3 ) x x x
Using the formula before the equation equals 0 0 0 For 3 x 2 = 5 − 4 x 3x^2 = 5 - 4x 3 x 2 = 5 − 4 x 3 x 2 + 4 x − 5 = 0 3x^2 + 4x - 5 = 0 3 x 2 + 4 x − 5 = 0 b b b c c c
Dropping the sign of b b b c c c In x 2 − 3 x − 10 = 0 x^2 - 3x - 10 = 0 x 2 − 3 x − 10 = 0 b = − 3 b = -3 b = − 3 c = − 10 c = -10 c = − 10 − b = 3 -b = 3 − b = 3 − 3 -3 − 3
Squaring a negative b b b If b = − 4 b = -4 b = − 4 b 2 = ( − 4 ) 2 = 16 b^2 = (-4)^2 = 16 b 2 = ( − 4 ) 2 = 16 ( − 4 ) 2 (-4)^2 ( − 4 ) 2 − 4 2 -4^2 − 4 2
Dividing only part of the top by 2 a 2a 2 a The whole numerator, − b ± b 2 − 4 a c -b \pm \sqrt{b^2 - 4ac} − b ± b 2 − 4 a c 2 a 2a 2 a
Rounding too early. Keep the full calculator value of the square root until the last step, then round.
Writing “no solution” too quickly. A negative b 2 − 4 a c b^2 - 4ac b 2 − 4 a c real roots. Double-check the signs in b 2 − 4 a c b^2 - 4ac b 2 − 4 a c
1. (Warm-up) State the values of a a a b b b c c c
(a) 4 x 2 − x + 7 = 0 4x^2 - x + 7 = 0 4 x 2 − x + 7 = 0
(b) 2 x 2 = 9 − 3 x 2x^2 = 9 - 3x 2 x 2 = 9 − 3 x
(c) x 2 − 5 = 0 x^2 - 5 = 0 x 2 − 5 = 0
Solution (a) a = 4 a = 4 a = 4 b = − 1 b = -1 b = − 1 c = 7 c = 7 c = 7
(b) Rearrange: 2 x 2 + 3 x − 9 = 0 2x^2 + 3x - 9 = 0 2 x 2 + 3 x − 9 = 0 a = 2 a = 2 a = 2 b = 3 b = 3 b = 3 c = − 9 c = -9 c = − 9
(c) There’s no x x x a = 1 a = 1 a = 1 b = 0 b = 0 b = 0 c = − 5 c = -5 c = − 5
2. (Warm-up) Solve x 2 + 6 x + 5 = 0 x^2 + 6x + 5 = 0 x 2 + 6 x + 5 = 0
Solution a = 1 a = 1 a = 1 b = 6 b = 6 b = 6 c = 5 c = 5 c = 5
x = − 6 ± 36 − 20 2 = − 6 ± 16 2 = − 6 ± 4 2 x = \frac{-6 \pm \sqrt{36 - 20}}{2} = \frac{-6 \pm \sqrt{16}}{2} = \frac{-6 \pm 4}{2} x = 2 − 6 ± 36 − 20 = 2 − 6 ± 16 = 2 − 6 ± 4 So x = − 1 x = -1 x = − 1 x = − 5 x = -5 x = − 5
3. (Core) Solve x 2 − 6 x + 4 = 0 x^2 - 6x + 4 = 0 x 2 − 6 x + 4 = 0
Solution a = 1 a = 1 a = 1 b = − 6 b = -6 b = − 6 c = 4 c = 4 c = 4
x = 6 ± 36 − 16 2 = 6 ± 20 2 x = \frac{6 \pm \sqrt{36 - 16}}{2} = \frac{6 \pm \sqrt{20}}{2} x = 2 6 ± 36 − 16 = 2 6 ± 20 Exact: x = 6 ± 20 2 x = \dfrac{6 \pm \sqrt{20}}{2} x = 2 6 ± 20 3 ± 5 3 \pm \sqrt{5} 3 ± 5
Decimals: x ≈ 5.24 x \approx 5.24 x ≈ 5.24 x ≈ 0.76 x \approx 0.76 x ≈ 0.76
4. (Core) Solve 2 x 2 + 3 x − 7 = 0 2x^2 + 3x - 7 = 0 2 x 2 + 3 x − 7 = 0
Solution a = 2 a = 2 a = 2 b = 3 b = 3 b = 3 c = − 7 c = -7 c = − 7
x = − 3 ± 9 − 4 ( 2 ) ( − 7 ) 4 = − 3 ± 9 + 56 4 = − 3 ± 65 4 x = \frac{-3 \pm \sqrt{9 - 4(2)(-7)}}{4} = \frac{-3 \pm \sqrt{9 + 56}}{4} = \frac{-3 \pm \sqrt{65}}{4} x = 4 − 3 ± 9 − 4 ( 2 ) ( − 7 ) = 4 − 3 ± 9 + 56 = 4 − 3 ± 65 With 65 ≈ 8.062 \sqrt{65} \approx 8.062 65 ≈ 8.062 x ≈ 5.062 4 ≈ 1.27 x \approx \dfrac{5.062}{4} \approx 1.27 x ≈ 4 5.062 ≈ 1.27 x ≈ − 11.062 4 ≈ − 2.77 x \approx \dfrac{-11.062}{4} \approx -2.77 x ≈ 4 − 11.062 ≈ − 2.77
5. (Core) Solve − x 2 + 4 x + 3 = 0 -x^2 + 4x + 3 = 0 − x 2 + 4 x + 3 = 0
Solution Multiplying both sides by − 1 -1 − 1 a a a
x 2 − 4 x − 3 = 0 x^2 - 4x - 3 = 0 x 2 − 4 x − 3 = 0 a = 1 a = 1 a = 1 b = − 4 b = -4 b = − 4 c = − 3 c = -3 c = − 3
x = 4 ± 16 + 12 2 = 4 ± 28 2 x = \frac{4 \pm \sqrt{16 + 12}}{2} = \frac{4 \pm \sqrt{28}}{2} x = 2 4 ± 16 + 12 = 2 4 ± 28 Exact: x = 4 ± 28 2 x = \dfrac{4 \pm \sqrt{28}}{2} x = 2 4 ± 28 2 ± 7 2 \pm \sqrt{7} 2 ± 7
Decimals: x ≈ 4.65 x \approx 4.65 x ≈ 4.65 x ≈ − 0.65 x \approx -0.65 x ≈ − 0.65
6. (Core) Choose a method for each equation, explain your choice, and solve. Round to two decimal places where needed.
(a) x 2 − 81 = 0 x^2 - 81 = 0 x 2 − 81 = 0
(b) x 2 + 5 x − 3 = 0 x^2 + 5x - 3 = 0 x 2 + 5 x − 3 = 0
Solution (a) Factoring: it’s a difference of squares. ( x − 9 ) ( x + 9 ) = 0 (x - 9)(x + 9) = 0 ( x − 9 ) ( x + 9 ) = 0 x = 9 x = 9 x = 9 x = − 9 x = -9 x = − 9
(b) No two integers multiply to − 3 -3 − 3 5 5 5 a = 1 a = 1 a = 1 b = 5 b = 5 b = 5 c = − 3 c = -3 c = − 3
x = − 5 ± 25 + 12 2 = − 5 ± 37 2 x = \frac{-5 \pm \sqrt{25 + 12}}{2} = \frac{-5 \pm \sqrt{37}}{2} x = 2 − 5 ± 25 + 12 = 2 − 5 ± 37 So x ≈ 0.54 x \approx 0.54 x ≈ 0.54 x ≈ − 5.54 x \approx -5.54 x ≈ − 5.54
7. (Core) Solve each equation, or explain why it has no real roots. Round to two decimal places.
(a) 5 x 2 − 2 x = 1 5x^2 - 2x = 1 5 x 2 − 2 x = 1
(b) 2 x 2 + 5 x + 4 = 0 2x^2 + 5x + 4 = 0 2 x 2 + 5 x + 4 = 0
Solution (a) Rearrange: 5 x 2 − 2 x − 1 = 0 5x^2 - 2x - 1 = 0 5 x 2 − 2 x − 1 = 0 a = 5 a = 5 a = 5 b = − 2 b = -2 b = − 2 c = − 1 c = -1 c = − 1
x = 2 ± 4 + 20 10 = 2 ± 24 10 x = \frac{2 \pm \sqrt{4 + 20}}{10} = \frac{2 \pm \sqrt{24}}{10} x = 10 2 ± 4 + 20 = 10 2 ± 24 So x ≈ 0.69 x \approx 0.69 x ≈ 0.69 x ≈ − 0.29 x \approx -0.29 x ≈ − 0.29
(b) b 2 − 4 a c = 25 − 4 ( 2 ) ( 4 ) = 25 − 32 = − 7 b^2 - 4ac = 25 - 4(2)(4) = 25 - 32 = -7 b 2 − 4 a c = 25 − 4 ( 2 ) ( 4 ) = 25 − 32 = − 7 y = 2 x 2 + 5 x + 4 y = 2x^2 + 5x + 4 y = 2 x 2 + 5 x + 4 x x x
8. (Challenge) Solve 0.5 x 2 − 1.2 x − 3 = 0 0.5x^2 - 1.2x - 3 = 0 0.5 x 2 − 1.2 x − 3 = 0
Solution a = 0.5 a = 0.5 a = 0.5 b = − 1.2 b = -1.2 b = − 1.2 c = − 3 c = -3 c = − 3
b 2 − 4 a c = ( − 1.2 ) 2 − 4 ( 0.5 ) ( − 3 ) = 1.44 + 6 = 7.44 b^2 - 4ac = (-1.2)^2 - 4(0.5)(-3) = 1.44 + 6 = 7.44 b 2 − 4 a c = ( − 1.2 ) 2 − 4 ( 0.5 ) ( − 3 ) = 1.44 + 6 = 7.44 x = 1.2 ± 7.44 2 ( 0.5 ) = 1.2 ± 7.44 1 x = \frac{1.2 \pm \sqrt{7.44}}{2(0.5)} = \frac{1.2 \pm \sqrt{7.44}}{1} x = 2 ( 0.5 ) 1.2 ± 7.44 = 1 1.2 ± 7.44 With 7.44 ≈ 2.728 \sqrt{7.44} \approx 2.728 7.44 ≈ 2.728 x ≈ 3.93 x \approx 3.93 x ≈ 3.93 x ≈ − 1.53 x \approx -1.53 x ≈ − 1.53
(You could also multiply the equation by 10 10 10 5 x 2 − 12 x − 30 = 0 5x^2 - 12x - 30 = 0 5 x 2 − 12 x − 30 = 0
9. (Challenge) Solve x 2 + 10 x + 7 = 0 x^2 + 10x + 7 = 0 x 2 + 10 x + 7 = 0
Solution Completing the square. Here a = 1 a = 1 a = 1
x 2 + 10 x = − 7 x 2 + 10 x + 25 = − 7 + 25 ( x + 5 ) 2 = 18 x + 5 = ± 18 x = − 5 ± 18 \begin{aligned}
x^2 + 10x &= -7 \\
x^2 + 10x + 25 &= -7 + 25 \\
(x + 5)^2 &= 18 \\
x + 5 &= \pm\sqrt{18} \\
x &= -5 \pm \sqrt{18}
\end{aligned} x 2 + 10 x x 2 + 10 x + 25 ( x + 5 ) 2 x + 5 x = − 7 = − 7 + 25 = 18 = ± 18 = − 5 ± 18 Formula. a = 1 a = 1 a = 1 b = 10 b = 10 b = 10 c = 7 c = 7 c = 7
x = − 10 ± 100 − 28 2 = − 10 ± 72 2 x = \frac{-10 \pm \sqrt{100 - 28}}{2} = \frac{-10 \pm \sqrt{72}}{2} x = 2 − 10 ± 100 − 28 = 2 − 10 ± 72 To compare, note that 72 = 4 × 18 = 2 18 \sqrt{72} = \sqrt{4 \times 18} = 2\sqrt{18} 72 = 4 × 18 = 2 18
x = − 10 ± 2 18 2 = − 5 ± 18 x = \frac{-10 \pm 2\sqrt{18}}{2} = -5 \pm \sqrt{18} x = 2 − 10 ± 2 18 = − 5 ± 18 The answers match: x ≈ − 0.76 x \approx -0.76 x ≈ − 0.76 x ≈ − 9.24 x \approx -9.24 x ≈ − 9.24
