Solve x^2=-63+16x

$$x ^ { 2 } = - 63 + 16 x$$

Solve for x

$$x=7$$

$$x=9$$

Steps Using Factoring

Subtract -63 from both sides.

$$x^{2}-\left(-63\right)=16x$$

The opposite of -63 is 63.

$$x^{2}+63=16x$$

Subtract 16x from both sides.

$$x^{2}+63-16x=0$$

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

$$x^{2}-16x+63=0$$

To solve the equation, factor x^{2}-16x+63 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.

$$a+b=-16 ab=63$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 63.

$$-1,-63 -3,-21 -7,-9$$

Calculate the sum for each pair.

$$-1-63=-64 -3-21=-24 -7-9=-16$$

The solution is the pair that gives sum -16.

$$a=-9 b=-7$$

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

$$\left(x-9\right)\left(x-7\right)$$

To find equation solutions, solve x-9=0 and x-7=0.

$$x=9 x=7$$

Steps Using Factoring By Grouping

Subtract -63 from both sides.

$$x^{2}-\left(-63\right)=16x$$

The opposite of -63 is 63.

$$x^{2}+63=16x$$

Subtract 16x from both sides.

$$x^{2}+63-16x=0$$

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

$$x^{2}-16x+63=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+63. To find a and b, set up a system to be solved.

$$a+b=-16 ab=1\times 63=63$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 63.

$$-1,-63 -3,-21 -7,-9$$

Calculate the sum for each pair.

$$-1-63=-64 -3-21=-24 -7-9=-16$$

The solution is the pair that gives sum -16.

$$a=-9 b=-7$$

Rewrite x^{2}-16x+63 as \left(x^{2}-9x\right)+\left(-7x+63\right).

$$\left(x^{2}-9x\right)+\left(-7x+63\right)$$

Factor out x in the first and -7 in the second group.

$$x\left(x-9\right)-7\left(x-9\right)$$

Factor out common term x-9 by using distributive property.

$$\left(x-9\right)\left(x-7\right)$$

To find equation solutions, solve x-9=0 and x-7=0.

$$x=9 x=7$$

Steps Using the Quadratic Formula

Subtract -63 from both sides.

$$x^{2}-\left(-63\right)=16x$$

The opposite of -63 is 63.

$$x^{2}+63=16x$$

Subtract 16x from both sides.

$$x^{2}+63-16x=0$$

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.

$$x^{2}-16x+63=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16 for b, and 63 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 63}}{2}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 63}}{2}$$

Multiply -4 times 63.

$$x=\frac{-\left(-16\right)±\sqrt{256-252}}{2}$$

Add 256 to -252.

$$x=\frac{-\left(-16\right)±\sqrt{4}}{2}$$

Take the square root of 4.

$$x=\frac{-\left(-16\right)±2}{2}$$

The opposite of -16 is 16.

$$x=\frac{16±2}{2}$$

Now solve the equation x=\frac{16±2}{2} when ± is plus. Add 16 to 2.

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

Divide 18 by 2.

$$x=9$$

Now solve the equation x=\frac{16±2}{2} when ± is minus. Subtract 2 from 16.

$$x=\frac{14}{2}$$

Divide 14 by 2.

$$x=7$$

The equation is now solved.

$$x=9 x=7$$

Steps for Completing the Square

Subtract 16x from both sides.

$$x^{2}-16x=-63$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-16x+\left(-8\right)^{2}=-63+\left(-8\right)^{2}$$

Square -8.

$$x^{2}-16x+64=-63+64$$

Add -63 to 64.

$$x^{2}-16x+64=1$$

Factor x^{2}-16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-8\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{1}$$

Simplify.

$$x-8=1 x-8=-1$$

Add 8 to both sides of the equation.

$$x=9 x=7$$