Solve g(y)=-y^2+2y+8.text{Find}g(-3)

$$g ( y ) = - y ^ { 2 } + 2 y + 8 . \text { Find } g ( - 3 )$$

Factor

$$-\left(y-4\right)\left(y+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -y^{2}+ay+by+8. To find a and b, set up a system to be solved.

$$a+b=2 ab=-8=-8$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -8.

$$-1,8 -2,4$$

Calculate the sum for each pair.

$$-1+8=7 -2+4=2$$

The solution is the pair that gives sum 2.

$$a=4 b=-2$$

Rewrite -y^{2}+2y+8 as \left(-y^{2}+4y\right)+\left(-2y+8\right).

$$\left(-y^{2}+4y\right)+\left(-2y+8\right)$$

Factor out -y in the first and -2 in the second group.

$$-y\left(y-4\right)-2\left(y-4\right)$$

Factor out common term y-4 by using distributive property.

$$\left(y-4\right)\left(-y-2\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-y^{2}+2y+8=0$$

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.

$$y=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Square 2.

$$y=\frac{-2±\sqrt{4-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$y=\frac{-2±\sqrt{4+4\times 8}}{2\left(-1\right)}$$

Multiply 4 times 8.

$$y=\frac{-2±\sqrt{4+32}}{2\left(-1\right)}$$

Add 4 to 32.

$$y=\frac{-2±\sqrt{36}}{2\left(-1\right)}$$

Take the square root of 36.

$$y=\frac{-2±6}{2\left(-1\right)}$$

Multiply 2 times -1.

$$y=\frac{-2±6}{-2}$$

Now solve the equation y=\frac{-2±6}{-2} when ± is plus. Add -2 to 6.

$$y=\frac{4}{-2}$$

Divide 4 by -2.

$$y=-2$$

Now solve the equation y=\frac{-2±6}{-2} when ± is minus. Subtract 6 from -2.

$$y=-\frac{8}{-2}$$

Divide -8 by -2.

$$y=4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -2 for x_{1} and 4 for x_{2}.

$$-y^{2}+2y+8=-\left(y-\left(-2\right)\right)\left(y-4\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-y^{2}+2y+8=-\left(y+2\right)\left(y-4\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 -2x -8 = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = 2 rs = -8$$

Two numbers r and s sum up to 2 exactly when the average of the two numbers is \frac{1}{2}*2 = 1. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = 1 - u s = 1 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -8

$$(1 - u) (1 + u) = -8$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$1 - u^2 = -8$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -8-1 = -9$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = 9 u = \pm\sqrt{9} = \pm 3$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =1 - 3 = -2 s = 1 + 3 = 4$$

Evaluate

$$-\left(y-4\right)\left(y+2\right)$$