Solve (a)=9a^2+46a.text{solvefor}h(a)=-5

$$( a ) = 9 a ^ { 2 } + 46 a . \text { solve for } h ( a ) = - 5$$

Solve for a

$$a=-5$$

$$a=0$$

Steps Using Factoring

Subtract 9a^{2} from both sides.

$$a-9a^{2}=46a$$

Subtract 46a from both sides.

$$a-9a^{2}-46a=0$$

Combine a and -46a to get -45a.

$$-45a-9a^{2}=0$$

Factor out a.

$$a\left(-45-9a\right)=0$$

To find equation solutions, solve a=0 and -45-9a=0.

$$a=0 a=-5$$

Steps Using the Quadratic Formula

Subtract 9a^{2} from both sides.

$$a-9a^{2}=46a$$

Subtract 46a from both sides.

$$a-9a^{2}-46a=0$$

Combine a and -46a to get -45a.

$$-45a-9a^{2}=0$$

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.

$$-9a^{2}-45a=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, -45 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-45\right)±\sqrt{\left(-45\right)^{2}}}{2\left(-9\right)}$$

Take the square root of \left(-45\right)^{2}.

$$a=\frac{-\left(-45\right)±45}{2\left(-9\right)}$$

The opposite of -45 is 45.

$$a=\frac{45±45}{2\left(-9\right)}$$

Multiply 2 times -9.

$$a=\frac{45±45}{-18}$$

Now solve the equation a=\frac{45±45}{-18} when ± is plus. Add 45 to 45.

$$a=\frac{90}{-18}$$

Divide 90 by -18.

$$a=-5$$

Now solve the equation a=\frac{45±45}{-18} when ± is minus. Subtract 45 from 45.

$$a=\frac{0}{-18}$$

Divide 0 by -18.

$$a=0$$

The equation is now solved.

$$a=-5 a=0$$

Steps for Completing the Square

Subtract 9a^{2} from both sides.

$$a-9a^{2}=46a$$

Subtract 46a from both sides.

$$a-9a^{2}-46a=0$$

Combine a and -46a to get -45a.

$$-45a-9a^{2}=0$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-9a^{2}-45a=0$$

Divide both sides by -9.

$$\frac{-9a^{2}-45a}{-9}=\frac{0}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$a^{2}+\left(-\frac{45}{-9}\right)a=\frac{0}{-9}$$

Divide -45 by -9.

$$a^{2}+5a=\frac{0}{-9}$$

Divide 0 by -9.

$$a^{2}+5a=0$$

Divide 5, the coefficient of the x term, by 2 to get \frac{5}{2}. Then add the square of \frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+5a+\left(\frac{5}{2}\right)^{2}=\left(\frac{5}{2}\right)^{2}$$

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+5a+\frac{25}{4}=\frac{25}{4}$$

Factor a^{2}+5a+\frac{25}{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}.

$$\left(a+\frac{5}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{5}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$a+\frac{5}{2}=\frac{5}{2} a+\frac{5}{2}=-\frac{5}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$a=0 a=-5$$