Solve 4x^2+25x-21

$$4 x ^ { 2 } + 25 x - 21$$

Factor

$$\left(4x-3\right)\left(x+7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx-21. To find a and b, set up a system to be solved.

$$a+b=25 ab=4\left(-21\right)=-84$$

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 -84.

$$-1,84 -2,42 -3,28 -4,21 -6,14 -7,12$$

Calculate the sum for each pair.

$$-1+84=83 -2+42=40 -3+28=25 -4+21=17 -6+14=8 -7+12=5$$

The solution is the pair that gives sum 25.

$$a=-3 b=28$$

Rewrite 4x^{2}+25x-21 as \left(4x^{2}-3x\right)+\left(28x-21\right).

$$\left(4x^{2}-3x\right)+\left(28x-21\right)$$

Factor out x in the first and 7 in the second group.

$$x\left(4x-3\right)+7\left(4x-3\right)$$

Factor out common term 4x-3 by using distributive property.

$$\left(4x-3\right)\left(x+7\right)$$

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.

$$4x^{2}+25x-21=0$$

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.

$$x=\frac{-25±\sqrt{25^{2}-4\times 4\left(-21\right)}}{2\times 4}$$

Square 25.

$$x=\frac{-25±\sqrt{625-4\times 4\left(-21\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-25±\sqrt{625-16\left(-21\right)}}{2\times 4}$$

Multiply -16 times -21.

$$x=\frac{-25±\sqrt{625+336}}{2\times 4}$$

Add 625 to 336.

$$x=\frac{-25±\sqrt{961}}{2\times 4}$$

Take the square root of 961.

$$x=\frac{-25±31}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-25±31}{8}$$

Now solve the equation x=\frac{-25±31}{8} when ± is plus. Add -25 to 31.

$$x=\frac{6}{8}$$

Reduce the fraction \frac{6}{8} to lowest terms by extracting and canceling out 2.

$$x=\frac{3}{4}$$

Now solve the equation x=\frac{-25±31}{8} when ± is minus. Subtract 31 from -25.

$$x=-\frac{56}{8}$$

Divide -56 by 8.

$$x=-7$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3}{4} for x_{1} and -7 for x_{2}.

$$4x^{2}+25x-21=4\left(x-\frac{3}{4}\right)\left(x-\left(-7\right)\right)$$

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

$$4x^{2}+25x-21=4\left(x-\frac{3}{4}\right)\left(x+7\right)$$

Subtract \frac{3}{4} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$4x^{2}+25x-21=4\times \frac{4x-3}{4}\left(x+7\right)$$

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

$$4x^{2}+25x-21=\left(4x-3\right)\left(x+7\right)$$

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

$$x ^ 2 +\frac{25}{4}x -\frac{21}{4} = 0$$

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

$$r + s = -\frac{25}{4} rs = -\frac{21}{4}$$

Two numbers r and s sum up to -\frac{25}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{25}{4} = -\frac{25}{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>

$$r = -\frac{25}{8} - u s = -\frac{25}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{21}{4}

$$(-\frac{25}{8} - u) (-\frac{25}{8} + u) = -\frac{21}{4}$$

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

$$\frac{625}{64} - u^2 = -\frac{21}{4}$$

Simplify the expression by subtracting \frac{625}{64} on both sides

$$-u^2 = -\frac{21}{4}-\frac{625}{64} = -\frac{961}{64}$$

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

$$u^2 = \frac{961}{64} u = \pm\sqrt{\frac{961}{64}} = \pm \frac{31}{8}$$

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

$$r =-\frac{25}{8} - \frac{31}{8} = -7 s = -\frac{25}{8} + \frac{31}{8} = 0.750$$

Evaluate

$$\left(4x-3\right)\left(x+7\right)$$