Solve r^2=8-7r

$$r ^ { 2 } = 8 - 7 r$$

Solve for r

$$r=-8$$

$$r=1$$

Steps Using Factoring

Subtract 8 from both sides.

$$r^{2}-8=-7r$$

Add 7r to both sides.

$$r^{2}-8+7r=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$r^{2}+7r-8=0$$

To solve the equation, factor r^{2}+7r-8 using formula r^{2}+\left(a+b\right)r+ab=\left(r+a\right)\left(r+b\right). To find a and b, set up a system to be solved.

$$a+b=7 ab=-8$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -8.

$$-1,8 -2,4$$

Calculate the sum for each pair.

$$-1+8=7 -2+4=2$$

The solution is the pair that gives sum 7.

$$a=-1 b=8$$

Rewrite factored expression \left(r+a\right)\left(r+b\right) using the obtained values.

$$\left(r-1\right)\left(r+8\right)$$

To find equation solutions, solve r-1=0 and r+8=0.

$$r=1 r=-8$$

Steps Using Factoring By Grouping

Subtract 8 from both sides.

$$r^{2}-8=-7r$$

Add 7r to both sides.

$$r^{2}-8+7r=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$r^{2}+7r-8=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as r^{2}+ar+br-8. To find a and b, set up a system to be solved.

$$a+b=7 ab=1\left(-8\right)=-8$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -8.

$$-1,8 -2,4$$

Calculate the sum for each pair.

$$-1+8=7 -2+4=2$$

The solution is the pair that gives sum 7.

$$a=-1 b=8$$

Rewrite r^{2}+7r-8 as \left(r^{2}-r\right)+\left(8r-8\right).

$$\left(r^{2}-r\right)+\left(8r-8\right)$$

Factor out r in the first and 8 in the second group.

$$r\left(r-1\right)+8\left(r-1\right)$$

Factor out common term r-1 by using distributive property.

$$\left(r-1\right)\left(r+8\right)$$

To find equation solutions, solve r-1=0 and r+8=0.

$$r=1 r=-8$$

Steps Using the Quadratic Formula

Subtract 8 from both sides.

$$r^{2}-8=-7r$$

Add 7r to both sides.

$$r^{2}-8+7r=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$r^{2}+7r-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 7 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-7±\sqrt{7^{2}-4\left(-8\right)}}{2}$$

Square 7.

$$r=\frac{-7±\sqrt{49-4\left(-8\right)}}{2}$$

Multiply -4 times -8.

$$r=\frac{-7±\sqrt{49+32}}{2}$$

Add 49 to 32.

$$r=\frac{-7±\sqrt{81}}{2}$$

Take the square root of 81.

$$r=\frac{-7±9}{2}$$

Now solve the equation r=\frac{-7±9}{2} when ± is plus. Add -7 to 9.

$$r=\frac{2}{2}$$

Divide 2 by 2.

$$r=1$$

Now solve the equation r=\frac{-7±9}{2} when ± is minus. Subtract 9 from -7.

$$r=-\frac{16}{2}$$

Divide -16 by 2.

$$r=-8$$

The equation is now solved.

$$r=1 r=-8$$

Steps for Completing the Square

Add 7r to both sides.

$$r^{2}+7r=8$$

Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. Then add the square of \frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}+7r+\left(\frac{7}{2}\right)^{2}=8+\left(\frac{7}{2}\right)^{2}$$

Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$r^{2}+7r+\frac{49}{4}=8+\frac{49}{4}$$

Add 8 to \frac{49}{4}.

$$r^{2}+7r+\frac{49}{4}=\frac{81}{4}$$

Factor r^{2}+7r+\frac{49}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(r+\frac{7}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r+\frac{7}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

$$r+\frac{7}{2}=\frac{9}{2} r+\frac{7}{2}=-\frac{9}{2}$$

Subtract \frac{7}{2} from both sides of the equation.

$$r=1 r=-8$$