Solve -4x^2+7x+2

$$-4 { x }^{ 2 } +7x+2$$

Factor

$$\left(2-x\right)\left(4x+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -4x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=7 ab=-4\times 2=-8$$

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 -8.

$$-1,8 -2,4$$

Calculate the sum for each pair.

$$-1+8=7 -2+4=2$$

The solution is the pair that gives sum 7.

$$a=8 b=-1$$

Rewrite -4x^{2}+7x+2 as \left(-4x^{2}+8x\right)+\left(-x+2\right).

$$\left(-4x^{2}+8x\right)+\left(-x+2\right)$$

Factor out 4x in -4x^{2}+8x.

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

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

$$\left(-x+2\right)\left(4x+1\right)$$

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.

$$-4x^{2}+7x+2=0$$

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.

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

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-4\right)\times 2}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-7±\sqrt{49+16\times 2}}{2\left(-4\right)}$$

Multiply 16 times 2.

$$x=\frac{-7±\sqrt{49+32}}{2\left(-4\right)}$$

Add 49 to 32.

$$x=\frac{-7±\sqrt{81}}{2\left(-4\right)}$$

Take the square root of 81.

$$x=\frac{-7±9}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-7±9}{-8}$$

Now solve the equation x=\frac{-7±9}{-8} when ± is plus. Add -7 to 9.

$$x=\frac{2}{-8}$$

Reduce the fraction \frac{2}{-8} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{4}$$

Now solve the equation x=\frac{-7±9}{-8} when ± is minus. Subtract 9 from -7.

$$x=-\frac{16}{-8}$$

Divide -16 by -8.

$$x=2$$

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

$$-4x^{2}+7x+2=-4\left(x-\left(-\frac{1}{4}\right)\right)\left(x-2\right)$$

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

$$-4x^{2}+7x+2=-4\left(x+\frac{1}{4}\right)\left(x-2\right)$$

Add \frac{1}{4} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-4x^{2}+7x+2=-4\times \frac{-4x-1}{-4}\left(x-2\right)$$

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

$$-4x^{2}+7x+2=\left(-4x-1\right)\left(x-2\right)$$

Evaluate

$$\left(2-x\right)\left(4x+1\right)$$