Solve 4x^2+7x-2

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

Factor

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

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.

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

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

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

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

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

Factor out common term 4x-1 by using distributive property.

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

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.

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

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.

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

Square 7.

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

Multiply -4 times 4.

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

Multiply -16 times -2.

$$x=\frac{-7±\sqrt{49+32}}{2\times 4}$$

Add 49 to 32.

$$x=\frac{-7±\sqrt{81}}{2\times 4}$$

Take the square root of 81.

$$x=\frac{-7±9}{2\times 4}$$

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}.

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

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

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

Subtract \frac{1}{4} from x by finding a common denominator and subtracting 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.

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

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 4

$$x ^ 2 +\frac{7}{4}x -\frac{1}{2} = 0$$

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

$$r + s = -\frac{7}{4} rs = -\frac{1}{2}$$

Two numbers r and s sum up to -\frac{7}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{7}{4} = -\frac{7}{8}. 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>

$$r = -\frac{7}{8} - u s = -\frac{7}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{2}

$$(-\frac{7}{8} - u) (-\frac{7}{8} + u) = -\frac{1}{2}$$

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

$$\frac{49}{64} - u^2 = -\frac{1}{2}$$

Simplify the expression by subtracting \frac{49}{64} on both sides

$$-u^2 = -\frac{1}{2}-\frac{49}{64} = -\frac{81}{64}$$

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

$$u^2 = \frac{81}{64} u = \pm\sqrt{\frac{81}{64}} = \pm \frac{9}{8}$$

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

$$r =-\frac{7}{8} - \frac{9}{8} = -2 s = -\frac{7}{8} + \frac{9}{8} = 0.250$$

Evaluate

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