Solve h^2+48=16h

$$h ^ { 2 } + 48 = 16 h$$

Solve for h

$$h=4$$

$$h=12$$

Steps Using Factoring

Subtract 16h from both sides.

$$h^{2}+48-16h=0$$

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

$$h^{2}-16h+48=0$$

To solve the equation, factor h^{2}-16h+48 using formula h^{2}+\left(a+b\right)h+ab=\left(h+a\right)\left(h+b\right). To find a and b, set up a system to be solved.

$$a+b=-16 ab=48$$

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 48.

$$-1,-48 -2,-24 -3,-16 -4,-12 -6,-8$$

Calculate the sum for each pair.

$$-1-48=-49 -2-24=-26 -3-16=-19 -4-12=-16 -6-8=-14$$

The solution is the pair that gives sum -16.

$$a=-12 b=-4$$

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

$$\left(h-12\right)\left(h-4\right)$$

To find equation solutions, solve h-12=0 and h-4=0.

$$h=12 h=4$$

Steps Using Factoring By Grouping

Subtract 16h from both sides.

$$h^{2}+48-16h=0$$

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

$$h^{2}-16h+48=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as h^{2}+ah+bh+48. To find a and b, set up a system to be solved.

$$a+b=-16 ab=1\times 48=48$$

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 48.

$$-1,-48 -2,-24 -3,-16 -4,-12 -6,-8$$

Calculate the sum for each pair.

$$-1-48=-49 -2-24=-26 -3-16=-19 -4-12=-16 -6-8=-14$$

The solution is the pair that gives sum -16.

$$a=-12 b=-4$$

Rewrite h^{2}-16h+48 as \left(h^{2}-12h\right)+\left(-4h+48\right).

$$\left(h^{2}-12h\right)+\left(-4h+48\right)$$

Factor out h in the first and -4 in the second group.

$$h\left(h-12\right)-4\left(h-12\right)$$

Factor out common term h-12 by using distributive property.

$$\left(h-12\right)\left(h-4\right)$$

To find equation solutions, solve h-12=0 and h-4=0.

$$h=12 h=4$$

Steps Using the Quadratic Formula

Subtract 16h from both sides.

$$h^{2}+48-16h=0$$

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.

$$h^{2}-16h+48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16 for b, and 48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$h=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 48}}{2}$$

Square -16.

$$h=\frac{-\left(-16\right)±\sqrt{256-4\times 48}}{2}$$

Multiply -4 times 48.

$$h=\frac{-\left(-16\right)±\sqrt{256-192}}{2}$$

Add 256 to -192.

$$h=\frac{-\left(-16\right)±\sqrt{64}}{2}$$

Take the square root of 64.

$$h=\frac{-\left(-16\right)±8}{2}$$

The opposite of -16 is 16.

$$h=\frac{16±8}{2}$$

Now solve the equation h=\frac{16±8}{2} when ± is plus. Add 16 to 8.

$$h=\frac{24}{2}$$

Divide 24 by 2.

$$h=12$$

Now solve the equation h=\frac{16±8}{2} when ± is minus. Subtract 8 from 16.

$$h=\frac{8}{2}$$

Divide 8 by 2.

$$h=4$$

The equation is now solved.

$$h=12 h=4$$

Steps for Completing the Square

Subtract 16h from both sides.

$$h^{2}+48-16h=0$$

Subtract 48 from both sides. Anything subtracted from zero gives its negation.

$$h^{2}-16h=-48$$

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.

$$h^{2}-16h+\left(-8\right)^{2}=-48+\left(-8\right)^{2}$$

Square -8.

$$h^{2}-16h+64=-48+64$$

Add -48 to 64.

$$h^{2}-16h+64=16$$

Factor h^{2}-16h+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}.

$$\left(h-8\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(h-8\right)^{2}}=\sqrt{16}$$

Simplify.

$$h-8=4 h-8=-4$$

Add 8 to both sides of the equation.

$$h=12 h=4$$