# Step-by-step Solution

## Solve the quadratic equation $-x^2+7x-10=0$

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### Videos

$x=5,\:x=2$

## Step-by-step explanation

Problem to solve:

$-x^2+7x-10=0$
1

Moving the term $-10$ to the other side of the equation with opposite sign

$-x^2+7x=10$
2

Factor the polynomial $-x^2+7x$ by it's GCF: $x$

$x\left(-x+7\right)=10$
3

Divide both sides of the equation by $x$

$-x+7=\frac{10}{x}$
4

Grouping terms

$-x+7+\frac{-10}{x}=0$
5

Moving the term $7$ to the other side of the equation with opposite sign

$-x+\frac{-10}{x}=-7$
6

Combine $-x+\frac{-10}{x}$ in a single fraction

$\frac{-10-x\cdot x}{x}=-7$
7

When multiplying two powers that have the same base ($x$), you can add the exponents

$\frac{-10-x^2}{x}=-7$
8

Multiply both sides of the equation by $x$

$-10-x^2=-7x$
9

We need to isolate the dependent variable $x$, we can do that by subtracting $-10$ from both sides of the equation

$-x^2=-7x+10$
10

Multiplying both sides of the equation by $-1$

$x^2=7x-10$
11

Grouping terms

$x^2-7x=-10$
12

Factor the polynomial $x^2-7x$ by it's GCF: $x$

$x\left(x-7\right)=-10$
13

Divide both sides of the equation by $x$

$x-7=\frac{-10}{x}$
14

Grouping terms

$x-7+\frac{10}{x}=0$
15

Moving the term $-7$ to the other side of the equation with opposite sign

$x+\frac{10}{x}=7$
16

Combine $x+\frac{10}{x}$ in a single fraction

$\frac{10+x\cdot x}{x}=7$
17

When multiplying two powers that have the same base ($x$), you can add the exponents

$\frac{10+x^2}{x}=7$
18

Multiply both sides of the equation by $x$

$10+x^2=7x$
19

Grouping terms

$10+x^2-7x=0$
20

To find the roots of a polynomial of the form $ax^2+bx+c$ we use the quadratic formula, where in this case $a=1$, $b=-7$ and $c=10$. Then substitute the values of the coefficients of the equation in the quadratic formula:

• $\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$x=\frac{7\pm 3}{2}$
21

To obtain the two solutions, divide the equation in two equations, one when $\pm$ is positive ($+$), and another when $\pm$ is negative ($-$)

$x=\frac{7+3}{2},\:x=\frac{7-3}{2}$
22

Subtract the values $7$ and $-3$

$x=\frac{7+3}{2},\:x=\frac{4}{2}$
23

Add the values $7$ and $3$

$x=\frac{10}{2},\:x=\frac{4}{2}$
24

Divide $10$ by $2$

$x=5,\:x=\frac{4}{2}$
25

Divide $4$ by $2$

$x=5,\:x=2$

$x=5,\:x=2$
$-x^2+7x-10=0$