Solve 15+14x-8x^2

$$15 + 14 x - 8 x ^ { 2 }$$

Factor

$$-\left(2x-5\right)\left(4x+3\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-8x^{2}+14x+15$$

Factor the expression by grouping. First, the expression needs to be rewritten as -8x^{2}+ax+bx+15. To find a and b, set up a system to be solved.

$$a+b=14 ab=-8\times 15=-120$$

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 -120.

$$-1,120 -2,60 -3,40 -4,30 -5,24 -6,20 -8,15 -10,12$$

Calculate the sum for each pair.

$$-1+120=119 -2+60=58 -3+40=37 -4+30=26 -5+24=19 -6+20=14 -8+15=7 -10+12=2$$

The solution is the pair that gives sum 14.

$$a=20 b=-6$$

Rewrite -8x^{2}+14x+15 as \left(-8x^{2}+20x\right)+\left(-6x+15\right).

$$\left(-8x^{2}+20x\right)+\left(-6x+15\right)$$

Factor out -4x in the first and -3 in the second group.

$$-4x\left(2x-5\right)-3\left(2x-5\right)$$

Factor out common term 2x-5 by using distributive property.

$$\left(2x-5\right)\left(-4x-3\right)$$

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.

$$-8x^{2}+14x+15=0$$

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.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-8\right)\times 15}}{2\left(-8\right)}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-8\right)\times 15}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-14±\sqrt{196+32\times 15}}{2\left(-8\right)}$$

Multiply 32 times 15.

$$x=\frac{-14±\sqrt{196+480}}{2\left(-8\right)}$$

Add 196 to 480.

$$x=\frac{-14±\sqrt{676}}{2\left(-8\right)}$$

Take the square root of 676.

$$x=\frac{-14±26}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-14±26}{-16}$$

Now solve the equation x=\frac{-14±26}{-16} when ± is plus. Add -14 to 26.

$$x=\frac{12}{-16}$$

Reduce the fraction \frac{12}{-16} to lowest terms by extracting and canceling out 4.

$$x=-\frac{3}{4}$$

Now solve the equation x=\frac{-14±26}{-16} when ± is minus. Subtract 26 from -14.

$$x=-\frac{40}{-16}$$

Reduce the fraction \frac{-40}{-16} to lowest terms by extracting and canceling out 8.

$$x=\frac{5}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{3}{4} for x_{1} and \frac{5}{2} for x_{2}.

$$-8x^{2}+14x+15=-8\left(x-\left(-\frac{3}{4}\right)\right)\left(x-\frac{5}{2}\right)$$

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

$$-8x^{2}+14x+15=-8\left(x+\frac{3}{4}\right)\left(x-\frac{5}{2}\right)$$

Add \frac{3}{4} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-8x^{2}+14x+15=-8\times \frac{-4x-3}{-4}\left(x-\frac{5}{2}\right)$$

Subtract \frac{5}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-8x^{2}+14x+15=-8\times \frac{-4x-3}{-4}\times \frac{-2x+5}{-2}$$

Multiply \frac{-4x-3}{-4} times \frac{-2x+5}{-2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-8x^{2}+14x+15=-8\times \frac{\left(-4x-3\right)\left(-2x+5\right)}{-4\left(-2\right)}$$

Multiply -4 times -2.

$$-8x^{2}+14x+15=-8\times \frac{\left(-4x-3\right)\left(-2x+5\right)}{8}$$

Cancel out 8, the greatest common factor in -8 and 8.

$$-8x^{2}+14x+15=-\left(-4x-3\right)\left(-2x+5\right)$$

Evaluate

$$15+14x-8x^{2}$$