Solve x(y)=-16y^2+148y-252

$$x(y)=-16 { y }^{ 2 } +148y-252$$

Factor

$$4\left(7-y\right)\left(4y-9\right)$$

Solution Steps

Factor out 4.

$$4\left(-4y^{2}+37y-63\right)$$

Consider -4y^{2}+37y-63. Factor the expression by grouping. First, the expression needs to be rewritten as -4y^{2}+ay+by-63. To find a and b, set up a system to be solved.

$$a+b=37 ab=-4\left(-63\right)=252$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 252.

$$1,252 2,126 3,84 4,63 6,42 7,36 9,28 12,21 14,18$$

Calculate the sum for each pair.

$$1+252=253 2+126=128 3+84=87 4+63=67 6+42=48 7+36=43 9+28=37 12+21=33 14+18=32$$

The solution is the pair that gives sum 37.

$$a=28 b=9$$

Rewrite -4y^{2}+37y-63 as \left(-4y^{2}+28y\right)+\left(9y-63\right).

$$\left(-4y^{2}+28y\right)+\left(9y-63\right)$$

Factor out 4y in the first and -9 in the second group.

$$4y\left(-y+7\right)-9\left(-y+7\right)$$

Factor out common term -y+7 by using distributive property.

$$\left(-y+7\right)\left(4y-9\right)$$

Rewrite the complete factored expression.

$$4\left(-y+7\right)\left(4y-9\right)$$

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.

$$-16y^{2}+148y-252=0$$

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.

$$y=\frac{-148±\sqrt{148^{2}-4\left(-16\right)\left(-252\right)}}{2\left(-16\right)}$$

Square 148.

$$y=\frac{-148±\sqrt{21904-4\left(-16\right)\left(-252\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$y=\frac{-148±\sqrt{21904+64\left(-252\right)}}{2\left(-16\right)}$$

Multiply 64 times -252.

$$y=\frac{-148±\sqrt{21904-16128}}{2\left(-16\right)}$$

Add 21904 to -16128.

$$y=\frac{-148±\sqrt{5776}}{2\left(-16\right)}$$

Take the square root of 5776.

$$y=\frac{-148±76}{2\left(-16\right)}$$

Multiply 2 times -16.

$$y=\frac{-148±76}{-32}$$

Now solve the equation y=\frac{-148±76}{-32} when ± is plus. Add -148 to 76.

$$y=-\frac{72}{-32}$$

Reduce the fraction \frac{-72}{-32} to lowest terms by extracting and canceling out 8.

$$y=\frac{9}{4}$$

Now solve the equation y=\frac{-148±76}{-32} when ± is minus. Subtract 76 from -148.

$$y=-\frac{224}{-32}$$

Divide -224 by -32.

$$y=7$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{9}{4} for x_{1} and 7 for x_{2}.

$$-16y^{2}+148y-252=-16\left(y-\frac{9}{4}\right)\left(y-7\right)$$

Subtract \frac{9}{4} from y by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-16y^{2}+148y-252=-16\times \frac{-4y+9}{-4}\left(y-7\right)$$

Cancel out 4, the greatest common factor in -16 and 4.

$$-16y^{2}+148y-252=4\left(-4y+9\right)\left(y-7\right)$$

Evaluate

$$-16y^{2}+148y-252$$