Solve x^2+7x-87=8x+3

$$x ^ { 2 } + 7 x - 87 = 8 x + 3$$

Solve for x

$$x=-9$$

$$x=10$$

Steps Using Factoring

Subtract 8x from both sides.

$$x^{2}+7x-87-8x=3$$

Combine 7x and -8x to get -x.

$$x^{2}-x-87=3$$

Subtract 3 from both sides.

$$x^{2}-x-87-3=0$$

Subtract 3 from -87 to get -90.

$$x^{2}-x-90=0$$

To solve the equation, factor x^{2}-x-90 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-1 ab=-90$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -90.

$$1,-90 2,-45 3,-30 5,-18 6,-15 9,-10$$

Calculate the sum for each pair.

$$1-90=-89 2-45=-43 3-30=-27 5-18=-13 6-15=-9 9-10=-1$$

The solution is the pair that gives sum -1.

$$a=-10 b=9$$

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

$$\left(x-10\right)\left(x+9\right)$$

To find equation solutions, solve x-10=0 and x+9=0.

$$x=10 x=-9$$

Steps Using Factoring By Grouping

Subtract 8x from both sides.

$$x^{2}+7x-87-8x=3$$

Combine 7x and -8x to get -x.

$$x^{2}-x-87=3$$

Subtract 3 from both sides.

$$x^{2}-x-87-3=0$$

Subtract 3 from -87 to get -90.

$$x^{2}-x-90=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-90. To find a and b, set up a system to be solved.

$$a+b=-1 ab=1\left(-90\right)=-90$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -90.

$$1,-90 2,-45 3,-30 5,-18 6,-15 9,-10$$

Calculate the sum for each pair.

$$1-90=-89 2-45=-43 3-30=-27 5-18=-13 6-15=-9 9-10=-1$$

The solution is the pair that gives sum -1.

$$a=-10 b=9$$

Rewrite x^{2}-x-90 as \left(x^{2}-10x\right)+\left(9x-90\right).

$$\left(x^{2}-10x\right)+\left(9x-90\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(x-10\right)+9\left(x-10\right)$$

Factor out common term x-10 by using distributive property.

$$\left(x-10\right)\left(x+9\right)$$

To find equation solutions, solve x-10=0 and x+9=0.

$$x=10 x=-9$$

Steps Using the Quadratic Formula

Subtract 8x from both sides.

$$x^{2}+7x-87-8x=3$$

Combine 7x and -8x to get -x.

$$x^{2}-x-87=3$$

Subtract 3 from both sides.

$$x^{2}-x-87-3=0$$

Subtract 3 from -87 to get -90.

$$x^{2}-x-90=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -90 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-90\right)}}{2}$$

Multiply -4 times -90.

$$x=\frac{-\left(-1\right)±\sqrt{1+360}}{2}$$

Add 1 to 360.

$$x=\frac{-\left(-1\right)±\sqrt{361}}{2}$$

Take the square root of 361.

$$x=\frac{-\left(-1\right)±19}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±19}{2}$$

Now solve the equation x=\frac{1±19}{2} when ± is plus. Add 1 to 19.

$$x=\frac{20}{2}$$

Divide 20 by 2.

$$x=10$$

Now solve the equation x=\frac{1±19}{2} when ± is minus. Subtract 19 from 1.

$$x=-\frac{18}{2}$$

Divide -18 by 2.

$$x=-9$$

The equation is now solved.

$$x=10 x=-9$$

Steps for Completing the Square

Subtract 8x from both sides.

$$x^{2}+7x-87-8x=3$$

Combine 7x and -8x to get -x.

$$x^{2}-x-87=3$$

Add 87 to both sides.

$$x^{2}-x=3+87$$

Add 3 and 87 to get 90.

$$x^{2}-x=90$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=90+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=90+\frac{1}{4}$$

Add 90 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{361}{4}$$

Factor x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{361}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{361}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{19}{2} x-\frac{1}{2}=-\frac{19}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=10 x=-9$$