Solve f(-3)=a(-3)^2+a^2-3+18=6

$$f ( - 3 ) = a ( - 3 ) ^ { 2 } + a ^ { 2 } - 3 + 18 = 6$$

Solve for a

$$a=\frac{3\sqrt{5}-9}{2}\approx -1.145898034$$

$$a=\frac{-3\sqrt{5}-9}{2}\approx -7.854101966$$

Steps Using the Quadratic Formula

Calculate -3 to the power of 2 and get 9.

$$a\times 9+a^{2}-3+18=6$$

Add -3 and 18 to get 15.

$$a\times 9+a^{2}+15=6$$

Subtract 6 from both sides.

$$a\times 9+a^{2}+15-6=0$$

Subtract 6 from 15 to get 9.

$$a\times 9+a^{2}+9=0$$

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.

$$a^{2}+9a+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 9 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-9±\sqrt{9^{2}-4\times 9}}{2}$$

Square 9.

$$a=\frac{-9±\sqrt{81-4\times 9}}{2}$$

Multiply -4 times 9.

$$a=\frac{-9±\sqrt{81-36}}{2}$$

Add 81 to -36.

$$a=\frac{-9±\sqrt{45}}{2}$$

Take the square root of 45.

$$a=\frac{-9±3\sqrt{5}}{2}$$

Now solve the equation a=\frac{-9±3\sqrt{5}}{2} when ± is plus. Add -9 to 3\sqrt{5}.

$$a=\frac{3\sqrt{5}-9}{2}$$

Now solve the equation a=\frac{-9±3\sqrt{5}}{2} when ± is minus. Subtract 3\sqrt{5} from -9.

$$a=\frac{-3\sqrt{5}-9}{2}$$

The equation is now solved.

$$a=\frac{3\sqrt{5}-9}{2} a=\frac{-3\sqrt{5}-9}{2}$$

Steps for Completing the Square

Calculate -3 to the power of 2 and get 9.

$$a\times 9+a^{2}-3+18=6$$

Add -3 and 18 to get 15.

$$a\times 9+a^{2}+15=6$$

Subtract 15 from both sides.

$$a\times 9+a^{2}=6-15$$

Subtract 15 from 6 to get -9.

$$a\times 9+a^{2}=-9$$

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.

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

Divide 9, the coefficient of the x term, by 2 to get \frac{9}{2}. Then add the square of \frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+9a+\frac{81}{4}=-9+\frac{81}{4}$$

Add -9 to \frac{81}{4}.

$$a^{2}+9a+\frac{81}{4}=\frac{45}{4}$$

Factor a^{2}+9a+\frac{81}{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}.

$$\left(a+\frac{9}{2}\right)^{2}=\frac{45}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{9}{2}\right)^{2}}=\sqrt{\frac{45}{4}}$$

Simplify.

$$a+\frac{9}{2}=\frac{3\sqrt{5}}{2} a+\frac{9}{2}=-\frac{3\sqrt{5}}{2}$$

Subtract \frac{9}{2} from both sides of the equation.

$$a=\frac{3\sqrt{5}-9}{2} a=\frac{-3\sqrt{5}-9}{2}$$