Solve 9x-4-2x^2

$$9x-4-2 { x }^{ 2 }$$

Factor

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

Solution Steps

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

$$-2x^{2}+9x-4$$

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

$$a+b=9 ab=-2\left(-4\right)=8$$

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 8.

$$1,8 2,4$$

Calculate the sum for each pair.

$$1+8=9 2+4=6$$

The solution is the pair that gives sum 9.

$$a=8 b=1$$

Rewrite -2x^{2}+9x-4 as \left(-2x^{2}+8x\right)+\left(x-4\right).

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

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

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

Factor out common term -x+4 by using distributive property.

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

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.

$$-2x^{2}+9x-4=0$$

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.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-2\right)\left(-4\right)}}{2\left(-2\right)}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-2\right)\left(-4\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-9±\sqrt{81+8\left(-4\right)}}{2\left(-2\right)}$$

Multiply 8 times -4.

$$x=\frac{-9±\sqrt{81-32}}{2\left(-2\right)}$$

Add 81 to -32.

$$x=\frac{-9±\sqrt{49}}{2\left(-2\right)}$$

Take the square root of 49.

$$x=\frac{-9±7}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-9±7}{-4}$$

Now solve the equation x=\frac{-9±7}{-4} when ± is plus. Add -9 to 7.

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

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

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

Now solve the equation x=\frac{-9±7}{-4} when ± is minus. Subtract 7 from -9.

$$x=-\frac{16}{-4}$$

Divide -16 by -4.

$$x=4$$

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

$$-2x^{2}+9x-4=-2\left(x-\frac{1}{2}\right)\left(x-4\right)$$

Subtract \frac{1}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-2x^{2}+9x-4=-2\times \frac{-2x+1}{-2}\left(x-4\right)$$

Cancel out 2, the greatest common factor in -2 and 2.

$$-2x^{2}+9x-4=\left(-2x+1\right)\left(x-4\right)$$

Evaluate

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