Solve 8x-x^2-15

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

Factor

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

Solution Steps

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

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

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

$$a+b=8 ab=-\left(-15\right)=15$$

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 15.

$$1,15 3,5$$

Calculate the sum for each pair.

$$1+15=16 3+5=8$$

The solution is the pair that gives sum 8.

$$a=5 b=3$$

Rewrite -x^{2}+8x-15 as \left(-x^{2}+5x\right)+\left(3x-15\right).

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

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

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

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

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

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.

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

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.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-1\right)\left(-15\right)}}{2\left(-1\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-1\right)\left(-15\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-8±\sqrt{64+4\left(-15\right)}}{2\left(-1\right)}$$

Multiply 4 times -15.

$$x=\frac{-8±\sqrt{64-60}}{2\left(-1\right)}$$

Add 64 to -60.

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

Take the square root of 4.

$$x=\frac{-8±2}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-8±2}{-2}$$

Now solve the equation x=\frac{-8±2}{-2} when ± is plus. Add -8 to 2.

$$x=-\frac{6}{-2}$$

Divide -6 by -2.

$$x=3$$

Now solve the equation x=\frac{-8±2}{-2} when ± is minus. Subtract 2 from -8.

$$x=-\frac{10}{-2}$$

Divide -10 by -2.

$$x=5$$

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

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

Evaluate

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