Solve x² = 2x + 15
Quadratic equation, worked out line by line the way a teacher would write it.
Answer
| Solutions | x = −3, x = 5 |
Step-by-step solution
9 steps-
1 Givenx^{2} = 2 x + 15
Solve for x.
-
2 Move every term to the left side so the right side is 0
Factoring and the quadratic formula both work on an equation of the form … = 0. Subtracting the right side from both sides gets there without changing the solutions.
x^{2} - 2 x - 15 = 0 -
3 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 = -2 and c = -15, signs included, shows which method fits: factoring, taking a square root or the quadratic formula.
a = 1,\quad b = -2,\quad c = -15 -
4 Factor the trinomial: find two numbers whose product is c = -15 and whose sum is b = -2
The goal is to write x² + bx + c as (x + p)(x + q). Multiplying that out gives x² + (p + q)x + p·q, so p and q must multiply to c = -15 and add up to b = -2.
List the pairs of numbers whose product is -15, with their signs, and pick the pair whose sum is -2.
(-5) \cdot 3 = -15,\qquad (-5) + 3 = -2 -
5 Write the factored form\left(x - 5\right) \left(x + 3\right) = 0
-
6 Zero product property: a product is 0 only when one of its factors is 0
0 is the only number with this property: if a·b = 0, then a = 0 or b = 0. That is why the equation was first rearranged to … = 0 and factored: now each factor can be set to 0 on its own, giving a simpler equation for each.
x - 5 = 0\quad \text{or} \quad x + 3 = 0 -
7 Add 5 to 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 = 5Move the constant terms to the right.
-
8 Subtract 3 from both sidesx = -3
Move the constant terms to the right.
-
9 Check\begin{aligned}x = -3:\quad 9 = 9\quad\checkmark\\ x = 5:\quad 25 = 25\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| −3 | 9 | 9 | ✓ |
| 5 | 25 | 25 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.