Solve 15x^2+7x-4

$$15 x ^ { 2 } + 7 x - 4$$

Factor

$$\left(3x-1\right)\left(5x+4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 15x^{2}+ax+bx-4. To find a and b, set up a system to be solved.

$$a+b=7 ab=15\left(-4\right)=-60$$

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 -60.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Calculate the sum for each pair.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

The solution is the pair that gives sum 7.

$$a=-5 b=12$$

Rewrite 15x^{2}+7x-4 as \left(15x^{2}-5x\right)+\left(12x-4\right).

$$\left(15x^{2}-5x\right)+\left(12x-4\right)$$

Factor out 5x in the first and 4 in the second group.

$$5x\left(3x-1\right)+4\left(3x-1\right)$$

Factor out common term 3x-1 by using distributive property.

$$\left(3x-1\right)\left(5x+4\right)$$

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.

$$15x^{2}+7x-4=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\times 15\left(-4\right)}}{2\times 15}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 15\left(-4\right)}}{2\times 15}$$

Multiply -4 times 15.

$$x=\frac{-7±\sqrt{49-60\left(-4\right)}}{2\times 15}$$

Multiply -60 times -4.

$$x=\frac{-7±\sqrt{49+240}}{2\times 15}$$

Add 49 to 240.

$$x=\frac{-7±\sqrt{289}}{2\times 15}$$

Take the square root of 289.

$$x=\frac{-7±17}{2\times 15}$$

Multiply 2 times 15.

$$x=\frac{-7±17}{30}$$

Now solve the equation x=\frac{-7±17}{30} when ± is plus. Add -7 to 17.

$$x=\frac{10}{30}$$

Reduce the fraction \frac{10}{30} to lowest terms by extracting and canceling out 10.

$$x=\frac{1}{3}$$

Now solve the equation x=\frac{-7±17}{30} when ± is minus. Subtract 17 from -7.

$$x=-\frac{24}{30}$$

Reduce the fraction \frac{-24}{30} to lowest terms by extracting and canceling out 6.

$$x=-\frac{4}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{3} for x_{1} and -\frac{4}{5} for x_{2}.

$$15x^{2}+7x-4=15\left(x-\frac{1}{3}\right)\left(x-\left(-\frac{4}{5}\right)\right)$$

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

$$15x^{2}+7x-4=15\left(x-\frac{1}{3}\right)\left(x+\frac{4}{5}\right)$$

Subtract \frac{1}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$15x^{2}+7x-4=15\times \frac{3x-1}{3}\left(x+\frac{4}{5}\right)$$

Add \frac{4}{5} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$15x^{2}+7x-4=15\times \frac{3x-1}{3}\times \frac{5x+4}{5}$$

Multiply \frac{3x-1}{3} times \frac{5x+4}{5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$15x^{2}+7x-4=15\times \frac{\left(3x-1\right)\left(5x+4\right)}{3\times 5}$$

Multiply 3 times 5.

$$15x^{2}+7x-4=15\times \frac{\left(3x-1\right)\left(5x+4\right)}{15}$$

Cancel out 15, the greatest common factor in 15 and 15.

$$15x^{2}+7x-4=\left(3x-1\right)\left(5x+4\right)$$

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 15

$$x ^ 2 +\frac{7}{15}x -\frac{4}{15} = 0$$

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

$$r + s = -\frac{7}{15} rs = -\frac{4}{15}$$

Two numbers r and s sum up to -\frac{7}{15} exactly when the average of the two numbers is \frac{1}{2}*-\frac{7}{15} = -\frac{7}{30}. 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>

$$r = -\frac{7}{30} - u s = -\frac{7}{30} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{4}{15}

$$(-\frac{7}{30} - u) (-\frac{7}{30} + u) = -\frac{4}{15}$$

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

$$\frac{49}{900} - u^2 = -\frac{4}{15}$$

Simplify the expression by subtracting \frac{49}{900} on both sides

$$-u^2 = -\frac{4}{15}-\frac{49}{900} = -\frac{289}{900}$$

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

$$u^2 = \frac{289}{900} u = \pm\sqrt{\frac{289}{900}} = \pm \frac{17}{30}$$

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

$$r =-\frac{7}{30} - \frac{17}{30} = -0.800 s = -\frac{7}{30} + \frac{17}{30} = 0.333$$

Evaluate

$$\left(3x-1\right)\left(5x+4\right)$$