Solve 5x^2-2+3x

$$5 { x }^{ 2 } -2+3x$$

Factor

$$\left(5x-2\right)\left(x+1\right)$$

Solution Steps

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

$$5x^{2}+3x-2$$

Factor the expression by grouping. First, the expression needs to be rewritten as 5x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=3 ab=5\left(-2\right)=-10$$

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 -10.

$$-1,10 -2,5$$

Calculate the sum for each pair.

$$-1+10=9 -2+5=3$$

The solution is the pair that gives sum 3.

$$a=-2 b=5$$

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

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

Factor out x in 5x^{2}-2x.

$$x\left(5x-2\right)+5x-2$$

Factor out common term 5x-2 by using distributive property.

$$\left(5x-2\right)\left(x+1\right)$$

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.

$$5x^{2}+3x-2=0$$

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.

$$x=\frac{-3±\sqrt{3^{2}-4\times 5\left(-2\right)}}{2\times 5}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 5\left(-2\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-3±\sqrt{9-20\left(-2\right)}}{2\times 5}$$

Multiply -20 times -2.

$$x=\frac{-3±\sqrt{9+40}}{2\times 5}$$

Add 9 to 40.

$$x=\frac{-3±\sqrt{49}}{2\times 5}$$

Take the square root of 49.

$$x=\frac{-3±7}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-3±7}{10}$$

Now solve the equation x=\frac{-3±7}{10} when ± is plus. Add -3 to 7.

$$x=\frac{4}{10}$$

Reduce the fraction \frac{4}{10} to lowest terms by extracting and canceling out 2.

$$x=\frac{2}{5}$$

Now solve the equation x=\frac{-3±7}{10} when ± is minus. Subtract 7 from -3.

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

Divide -10 by 10.

$$x=-1$$

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

$$5x^{2}+3x-2=5\left(x-\frac{2}{5}\right)\left(x-\left(-1\right)\right)$$

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

$$5x^{2}+3x-2=5\left(x-\frac{2}{5}\right)\left(x+1\right)$$

Subtract \frac{2}{5} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$5x^{2}+3x-2=5\times \frac{5x-2}{5}\left(x+1\right)$$

Cancel out 5, the greatest common factor in 5 and 5.

$$5x^{2}+3x-2=\left(5x-2\right)\left(x+1\right)$$

Evaluate

$$\left(5x-2\right)\left(x+1\right)$$