Solve 7-15x=-2x^2

$$7 - 15 x = - 2 x ^ { 2 }$$

Solve for x

$$x=\frac{1}{2}=0.5$$

$$x=7$$

Steps Using Factoring By Grouping

Add 2x^{2} to both sides.

$$7-15x+2x^{2}=0$$

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

$$2x^{2}-15x+7=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2x^{2}+ax+bx+7. To find a and b, set up a system to be solved.

$$a+b=-15 ab=2\times 7=14$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 14.

$$-1,-14 -2,-7$$

Calculate the sum for each pair.

$$-1-14=-15 -2-7=-9$$

The solution is the pair that gives sum -15.

$$a=-14 b=-1$$

Rewrite 2x^{2}-15x+7 as \left(2x^{2}-14x\right)+\left(-x+7\right).

$$\left(2x^{2}-14x\right)+\left(-x+7\right)$$

Factor out 2x in the first and -1 in the second group.

$$2x\left(x-7\right)-\left(x-7\right)$$

Factor out common term x-7 by using distributive property.

$$\left(x-7\right)\left(2x-1\right)$$

To find equation solutions, solve x-7=0 and 2x-1=0.

$$x=7 x=\frac{1}{2}$$

Steps Using the Quadratic Formula

Add 2x^{2} to both sides.

$$7-15x+2x^{2}=0$$

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.

$$2x^{2}-15x+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -15 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 2\times 7}}{2\times 2}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 2\times 7}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-15\right)±\sqrt{225-8\times 7}}{2\times 2}$$

Multiply -8 times 7.

$$x=\frac{-\left(-15\right)±\sqrt{225-56}}{2\times 2}$$

Add 225 to -56.

$$x=\frac{-\left(-15\right)±\sqrt{169}}{2\times 2}$$

Take the square root of 169.

$$x=\frac{-\left(-15\right)±13}{2\times 2}$$

The opposite of -15 is 15.

$$x=\frac{15±13}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{15±13}{4}$$

Now solve the equation x=\frac{15±13}{4} when ± is plus. Add 15 to 13.

$$x=\frac{28}{4}$$

Divide 28 by 4.

$$x=7$$

Now solve the equation x=\frac{15±13}{4} when ± is minus. Subtract 13 from 15.

$$x=\frac{2}{4}$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{2}$$

The equation is now solved.

$$x=7 x=\frac{1}{2}$$

Steps for Completing the Square

Add 2x^{2} to both sides.

$$7-15x+2x^{2}=0$$

Subtract 7 from both sides. Anything subtracted from zero gives its negation.

$$-15x+2x^{2}=-7$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$2x^{2}-15x=-7$$

Divide both sides by 2.

$$\frac{2x^{2}-15x}{2}=-\frac{7}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{15}{2}x=-\frac{7}{2}$$

Divide -\frac{15}{2}, the coefficient of the x term, by 2 to get -\frac{15}{4}. Then add the square of -\frac{15}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{15}{2}x+\left(-\frac{15}{4}\right)^{2}=-\frac{7}{2}+\left(-\frac{15}{4}\right)^{2}$$

Square -\frac{15}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{15}{2}x+\frac{225}{16}=-\frac{7}{2}+\frac{225}{16}$$

Add -\frac{7}{2} to \frac{225}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{15}{2}x+\frac{225}{16}=\frac{169}{16}$$

Factor x^{2}-\frac{15}{2}x+\frac{225}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{15}{4}\right)^{2}=\frac{169}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{4}\right)^{2}}=\sqrt{\frac{169}{16}}$$

Simplify.

$$x-\frac{15}{4}=\frac{13}{4} x-\frac{15}{4}=-\frac{13}{4}$$

Add \frac{15}{4} to both sides of the equation.

$$x=7 x=\frac{1}{2}$$