Solve -14x^2+45x+14

$$-14 { x }^{ 2 } +45x+14$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -14x^{2}+ax+bx+14. To find a and b, set up a system to be solved.

$$a+b=45 ab=-14\times 14=-196$$

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 -196.

$$-1,196 -2,98 -4,49 -7,28 -14,14$$

Calculate the sum for each pair.

$$-1+196=195 -2+98=96 -4+49=45 -7+28=21 -14+14=0$$

The solution is the pair that gives sum 45.

$$a=49 b=-4$$

Rewrite -14x^{2}+45x+14 as \left(-14x^{2}+49x\right)+\left(-4x+14\right).

$$\left(-14x^{2}+49x\right)+\left(-4x+14\right)$$

Factor out -7x in the first and -2 in the second group.

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

Factor out common term 2x-7 by using distributive property.

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

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.

$$-14x^{2}+45x+14=0$$

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.

$$x=\frac{-45±\sqrt{45^{2}-4\left(-14\right)\times 14}}{2\left(-14\right)}$$

Square 45.

$$x=\frac{-45±\sqrt{2025-4\left(-14\right)\times 14}}{2\left(-14\right)}$$

Multiply -4 times -14.

$$x=\frac{-45±\sqrt{2025+56\times 14}}{2\left(-14\right)}$$

Multiply 56 times 14.

$$x=\frac{-45±\sqrt{2025+784}}{2\left(-14\right)}$$

Add 2025 to 784.

$$x=\frac{-45±\sqrt{2809}}{2\left(-14\right)}$$

Take the square root of 2809.

$$x=\frac{-45±53}{2\left(-14\right)}$$

Multiply 2 times -14.

$$x=\frac{-45±53}{-28}$$

Now solve the equation x=\frac{-45±53}{-28} when ± is plus. Add -45 to 53.

$$x=\frac{8}{-28}$$

Reduce the fraction \frac{8}{-28} to lowest terms by extracting and canceling out 4.

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

Now solve the equation x=\frac{-45±53}{-28} when ± is minus. Subtract 53 from -45.

$$x=-\frac{98}{-28}$$

Reduce the fraction \frac{-98}{-28} to lowest terms by extracting and canceling out 14.

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

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

$$-14x^{2}+45x+14=-14\left(x-\left(-\frac{2}{7}\right)\right)\left(x-\frac{7}{2}\right)$$

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

$$-14x^{2}+45x+14=-14\left(x+\frac{2}{7}\right)\left(x-\frac{7}{2}\right)$$

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

$$-14x^{2}+45x+14=-14\times \frac{-7x-2}{-7}\left(x-\frac{7}{2}\right)$$

Subtract \frac{7}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-14x^{2}+45x+14=-14\times \frac{-7x-2}{-7}\times \frac{-2x+7}{-2}$$

Multiply \frac{-7x-2}{-7} times \frac{-2x+7}{-2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-14x^{2}+45x+14=-14\times \frac{\left(-7x-2\right)\left(-2x+7\right)}{-7\left(-2\right)}$$

Multiply -7 times -2.

$$-14x^{2}+45x+14=-14\times \frac{\left(-7x-2\right)\left(-2x+7\right)}{14}$$

Cancel out 14, the greatest common factor in -14 and 14.

$$-14x^{2}+45x+14=-\left(-7x-2\right)\left(-2x+7\right)$$

Evaluate

$$14+45x-14x^{2}$$