Solve h(x)=2x^2+7x-4

$$h ( x ) = 2 x ^ { 2 } + 7 x - 4$$

Factor

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

Solution Steps

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.

$$a+b=7 ab=2\left(-4\right)=-8$$

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.

$$a=-1 b=8$$

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

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

Factor out x in the first and 4 in the second group.

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

Factor out common term 2x-1 by using distributive property.

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

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.

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

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.

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

Square 7.

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

Multiply -4 times 2.

$$x=\frac{-7±\sqrt{49-8\left(-4\right)}}{2\times 2}$$

Multiply -8 times -4.

$$x=\frac{-7±\sqrt{49+32}}{2\times 2}$$

Add 49 to 32.

$$x=\frac{-7±\sqrt{81}}{2\times 2}$$

Take the square root of 81.

$$x=\frac{-7±9}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-7±9}{4}$$

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

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

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{-7±9}{4} when ± is minus. Subtract 9 from -7.

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

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}.

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

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

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

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

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

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

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

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 2

$$x ^ 2 +\frac{7}{2}x -2 = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -\frac{7}{2} rs = -2$$

Two numbers r and s sum up to -\frac{7}{2} exactly when the average of the two numbers is \frac{1}{2}*-\frac{7}{2} = -\frac{7}{4}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -\frac{7}{4} - u s = -\frac{7}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -2

$$(-\frac{7}{4} - u) (-\frac{7}{4} + u) = -2$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{49}{16} - u^2 = -2$$

Simplify the expression by subtracting \frac{49}{16} on both sides

$$-u^2 = -2-\frac{49}{16} = -\frac{81}{16}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{81}{16} u = \pm\sqrt{\frac{81}{16}} = \pm \frac{9}{4}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-\frac{7}{4} - \frac{9}{4} = -4 s = -\frac{7}{4} + \frac{9}{4} = 0.500$$

Evaluate

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