Solve -8y^2-12y+5)+(7y^2-8

$$- 8 y ^ { 2 } - 12 y + 5 ) + ( 7 y ^ { 2 } - 8$$

Factor

$$-8\left(y-\frac{-\sqrt{19}-3}{4}\right)\left(y-\frac{\sqrt{19}-3}{4}\right)$$

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.

$$-8y^{2}-12y+5=0$$

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.

$$y=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-8\right)\times 5}}{2\left(-8\right)}$$

Square -12.

$$y=\frac{-\left(-12\right)±\sqrt{144-4\left(-8\right)\times 5}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$y=\frac{-\left(-12\right)±\sqrt{144+32\times 5}}{2\left(-8\right)}$$

Multiply 32 times 5.

$$y=\frac{-\left(-12\right)±\sqrt{144+160}}{2\left(-8\right)}$$

Add 144 to 160.

$$y=\frac{-\left(-12\right)±\sqrt{304}}{2\left(-8\right)}$$

Take the square root of 304.

$$y=\frac{-\left(-12\right)±4\sqrt{19}}{2\left(-8\right)}$$

The opposite of -12 is 12.

$$y=\frac{12±4\sqrt{19}}{2\left(-8\right)}$$

Multiply 2 times -8.

$$y=\frac{12±4\sqrt{19}}{-16}$$

Now solve the equation y=\frac{12±4\sqrt{19}}{-16} when ± is plus. Add 12 to 4\sqrt{19}.

$$y=\frac{4\sqrt{19}+12}{-16}$$

Divide 12+4\sqrt{19} by -16.

$$y=\frac{-\sqrt{19}-3}{4}$$

Now solve the equation y=\frac{12±4\sqrt{19}}{-16} when ± is minus. Subtract 4\sqrt{19} from 12.

$$y=\frac{12-4\sqrt{19}}{-16}$$

Divide 12-4\sqrt{19} by -16.

$$y=\frac{\sqrt{19}-3}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-3-\sqrt{19}}{4} for x_{1} and \frac{-3+\sqrt{19}}{4} for x_{2}.

$$-8y^{2}-12y+5=-8\left(y-\frac{-\sqrt{19}-3}{4}\right)\left(y-\frac{\sqrt{19}-3}{4}\right)$$

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.

$$x ^ 2 +\frac{3}{2}x -\frac{5}{8} = 0$$

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

$$r + s = -\frac{3}{2} rs = -\frac{5}{8}$$

Two numbers r and s sum up to -\frac{3}{2} exactly when the average of the two numbers is \frac{1}{2}*-\frac{3}{2} = -\frac{3}{4}. 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>

$$r = -\frac{3}{4} - u s = -\frac{3}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{5}{8}

$$(-\frac{3}{4} - u) (-\frac{3}{4} + u) = -\frac{5}{8}$$

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

$$\frac{9}{16} - u^2 = -\frac{5}{8}$$

Simplify the expression by subtracting \frac{9}{16} on both sides

$$-u^2 = -\frac{5}{8}-\frac{9}{16} = -\frac{19}{16}$$

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

$$u^2 = \frac{19}{16} u = \pm\sqrt{\frac{19}{16}} = \pm \frac{\sqrt{19}}{4}$$

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

$$r =-\frac{3}{4} - \frac{\sqrt{19}}{4} = -1.840 s = -\frac{3}{4} + \frac{\sqrt{19}}{4} = 0.340$$

Evaluate

$$5-12y-8y^{2}$$