Solve x² + 10x + 7 = 0 by completing the square
Quadratic equation, worked out line by line the way a teacher would write it.
Answer
| Solutions | x = −5 − 3√2, x = 3√2 − 5 |
Step-by-step solution
8 steps-
1 Givenx^{2} + 10 x + 7 = 0
Solve for x.
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2 This is a quadratic in standard form ax² + bx + c = 0
Every quadratic equation can be arranged as ax² + bx + c = 0. Reading off a = 1, b = 10 and c = 7, signs included, shows which method fits: factoring, taking a square root or the quadratic formula.
a = 1,\quad b = 10,\quad c = 7 -
3 Subtract 7 from both sides
An equation stays true as long as you do the same thing to both sides, like adding the same weight to both pans of a balance. Adding or subtracting the same amount on both sides moves a term to the other side, where it appears with the opposite sign.
x^{2} + 10 x = -7 -
4 Add (b/2)² = 25 to both sides to complete the square
(x + h)² = x² + 2hx + h². With h = 5, half of the x coefficient, adding h² = 25 makes the left side a perfect square, (x + h)². The same amount is added on the right to keep the equation balanced; then a square root finishes the job.
x^{2} + 10 x + 25 = 25 - 7Half the x coefficient is 5; its square is 25.
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5 The left side is now a perfect square\left(x + 5\right)^{2} = 18
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6 Take the square root of both sides — remember both signs
A number and its negative give the same result when raised to an even power: 3² = 9 and (−3)² = 9. So if something squared equals k, that something is √k or −√k. Forgetting the minus sign loses a solution.
x + 5 = \pm 3 \sqrt{2} -
7 Subtract 5 from both sidesx = -5 - 3 \sqrt{2}\quad \text{or} \quad x = 3 \sqrt{2} - 5
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8 Check\begin{aligned}x = -5 - 3 \sqrt{2}:\quad 0 = 0\quad\checkmark\\ x = 3 \sqrt{2} - 5:\quad 0 = 0\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| −5 − 3√2 | 0 | 0 | ✓ |
| 3√2 − 5 | 0 | 0 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.