Solve 4h^2+4h^2+3h+4=52

$$4 h ^ { 2 } + 4 h ^ { 2 } + 3 h + 4 = 52$$

Solve for h

$$h = \frac{\sqrt{1545} - 3}{16} \approx 2.269155501$$

$$h=\frac{-\sqrt{1545}-3}{16}\approx -2.644155501$$

Steps Using the Quadratic Formula

Combine 4h^{2} and 4h^{2} to get 8h^{2}.

$$8h^{2}+3h+4=52$$

Subtract 52 from both sides.

$$8h^{2}+3h+4-52=0$$

Subtract 52 from 4 to get -48.

$$8h^{2}+3h-48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 3 for b, and -48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$h=\frac{-3±\sqrt{3^{2}-4\times 8\left(-48\right)}}{2\times 8}$$

Square 3.

$$h=\frac{-3±\sqrt{9-4\times 8\left(-48\right)}}{2\times 8}$$

Multiply -4 times 8.

$$h=\frac{-3±\sqrt{9-32\left(-48\right)}}{2\times 8}$$

Multiply -32 times -48.

$$h=\frac{-3±\sqrt{9+1536}}{2\times 8}$$

Add 9 to 1536.

$$h=\frac{-3±\sqrt{1545}}{2\times 8}$$

Multiply 2 times 8.

$$h=\frac{-3±\sqrt{1545}}{16}$$

Now solve the equation h=\frac{-3±\sqrt{1545}}{16} when ± is plus. Add -3 to \sqrt{1545}.

$$h=\frac{\sqrt{1545}-3}{16}$$

Now solve the equation h=\frac{-3±\sqrt{1545}}{16} when ± is minus. Subtract \sqrt{1545} from -3.

$$h=\frac{-\sqrt{1545}-3}{16}$$

The equation is now solved.

$$h=\frac{\sqrt{1545}-3}{16} h=\frac{-\sqrt{1545}-3}{16}$$

Steps for Completing the Square

Combine 4h^{2} and 4h^{2} to get 8h^{2}.

$$8h^{2}+3h+4=52$$

Subtract 4 from both sides.

$$8h^{2}+3h=52-4$$

Subtract 4 from 52 to get 48.

$$8h^{2}+3h=48$$

Divide both sides by 8.

$$\frac{8h^{2}+3h}{8}=\frac{48}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$h^{2}+\frac{3}{8}h=\frac{48}{8}$$

Divide 48 by 8.

$$h^{2}+\frac{3}{8}h=6$$

Divide \frac{3}{8}, the coefficient of the x term, by 2 to get \frac{3}{16}. Then add the square of \frac{3}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$h^{2}+\frac{3}{8}h+\left(\frac{3}{16}\right)^{2}=6+\left(\frac{3}{16}\right)^{2}$$

Square \frac{3}{16} by squaring both the numerator and the denominator of the fraction.

$$h^{2}+\frac{3}{8}h+\frac{9}{256}=6+\frac{9}{256}$$

Add 6 to \frac{9}{256}.

$$h^{2}+\frac{3}{8}h+\frac{9}{256}=\frac{1545}{256}$$

Factor h^{2}+\frac{3}{8}h+\frac{9}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(h+\frac{3}{16}\right)^{2}=\frac{1545}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(h+\frac{3}{16}\right)^{2}}=\sqrt{\frac{1545}{256}}$$

Simplify.

$$h+\frac{3}{16}=\frac{\sqrt{1545}}{16} h+\frac{3}{16}=-\frac{\sqrt{1545}}{16}$$

Subtract \frac{3}{16} from both sides of the equation.

$$h=\frac{\sqrt{1545}-3}{16} h=\frac{-\sqrt{1545}-3}{16}$$