Solve 10x^2+13x-3=

$$10 x ^ { 2 } + 13 x - 3 =$$

Factor

$$\left(5x-1\right)\left(2x+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 10x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=13 ab=10\left(-3\right)=-30$$

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 -30.

$$-1,30 -2,15 -3,10 -5,6$$

Calculate the sum for each pair.

$$-1+30=29 -2+15=13 -3+10=7 -5+6=1$$

The solution is the pair that gives sum 13.

$$a=-2 b=15$$

Rewrite 10x^{2}+13x-3 as \left(10x^{2}-2x\right)+\left(15x-3\right).

$$\left(10x^{2}-2x\right)+\left(15x-3\right)$$

Factor out 2x in the first and 3 in the second group.

$$2x\left(5x-1\right)+3\left(5x-1\right)$$

Factor out common term 5x-1 by using distributive property.

$$\left(5x-1\right)\left(2x+3\right)$$

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.

$$10x^{2}+13x-3=0$$

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.

$$x=\frac{-13±\sqrt{13^{2}-4\times 10\left(-3\right)}}{2\times 10}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\times 10\left(-3\right)}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-13±\sqrt{169-40\left(-3\right)}}{2\times 10}$$

Multiply -40 times -3.

$$x=\frac{-13±\sqrt{169+120}}{2\times 10}$$

Add 169 to 120.

$$x=\frac{-13±\sqrt{289}}{2\times 10}$$

Take the square root of 289.

$$x=\frac{-13±17}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{-13±17}{20}$$

Now solve the equation x=\frac{-13±17}{20} when ± is plus. Add -13 to 17.

$$x=\frac{4}{20}$$

Reduce the fraction \frac{4}{20} to lowest terms by extracting and canceling out 4.

$$x=\frac{1}{5}$$

Now solve the equation x=\frac{-13±17}{20} when ± is minus. Subtract 17 from -13.

$$x=-\frac{30}{20}$$

Reduce the fraction \frac{-30}{20} to lowest terms by extracting and canceling out 10.

$$x=-\frac{3}{2}$$

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

$$10x^{2}+13x-3=10\left(x-\frac{1}{5}\right)\left(x-\left(-\frac{3}{2}\right)\right)$$

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

$$10x^{2}+13x-3=10\left(x-\frac{1}{5}\right)\left(x+\frac{3}{2}\right)$$

Subtract \frac{1}{5} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$10x^{2}+13x-3=10\times \frac{5x-1}{5}\left(x+\frac{3}{2}\right)$$

Add \frac{3}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$10x^{2}+13x-3=10\times \frac{5x-1}{5}\times \frac{2x+3}{2}$$

Multiply \frac{5x-1}{5} times \frac{2x+3}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$10x^{2}+13x-3=10\times \frac{\left(5x-1\right)\left(2x+3\right)}{5\times 2}$$

Multiply 5 times 2.

$$10x^{2}+13x-3=10\times \frac{\left(5x-1\right)\left(2x+3\right)}{10}$$

Cancel out 10, the greatest common factor in 10 and 10.

$$10x^{2}+13x-3=\left(5x-1\right)\left(2x+3\right)$$

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 10

$$x ^ 2 +\frac{13}{10}x -\frac{3}{10} = 0$$

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

$$r + s = -\frac{13}{10} rs = -\frac{3}{10}$$

Two numbers r and s sum up to -\frac{13}{10} exactly when the average of the two numbers is \frac{1}{2}*-\frac{13}{10} = -\frac{13}{20}. 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>

$$r = -\frac{13}{20} - u s = -\frac{13}{20} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{3}{10}

$$(-\frac{13}{20} - u) (-\frac{13}{20} + u) = -\frac{3}{10}$$

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

$$\frac{169}{400} - u^2 = -\frac{3}{10}$$

Simplify the expression by subtracting \frac{169}{400} on both sides

$$-u^2 = -\frac{3}{10}-\frac{169}{400} = -\frac{289}{400}$$

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

$$u^2 = \frac{289}{400} u = \pm\sqrt{\frac{289}{400}} = \pm \frac{17}{20}$$

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

$$r =-\frac{13}{20} - \frac{17}{20} = -1.500 s = -\frac{13}{20} + \frac{17}{20} = 0.200$$

Evaluate

$$\left(5x-1\right)\left(2x+3\right)$$