Solve 6p^2-7p-10

$$6 p ^ { 2 } - 7 p - 10$$

Factor

$$\left(p-2\right)\left(6p+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 6p^{2}+ap+bp-10. To find a and b, set up a system to be solved.

$$a+b=-7 ab=6\left(-10\right)=-60$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. 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.

$$a=-12 b=5$$

Rewrite 6p^{2}-7p-10 as \left(6p^{2}-12p\right)+\left(5p-10\right).

$$\left(6p^{2}-12p\right)+\left(5p-10\right)$$

Factor out 6p in the first and 5 in the second group.

$$6p\left(p-2\right)+5\left(p-2\right)$$

Factor out common term p-2 by using distributive property.

$$\left(p-2\right)\left(6p+5\right)$$

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.

$$6p^{2}-7p-10=0$$

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.

$$p=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 6\left(-10\right)}}{2\times 6}$$

Square -7.

$$p=\frac{-\left(-7\right)±\sqrt{49-4\times 6\left(-10\right)}}{2\times 6}$$

Multiply -4 times 6.

$$p=\frac{-\left(-7\right)±\sqrt{49-24\left(-10\right)}}{2\times 6}$$

Multiply -24 times -10.

$$p=\frac{-\left(-7\right)±\sqrt{49+240}}{2\times 6}$$

Add 49 to 240.

$$p=\frac{-\left(-7\right)±\sqrt{289}}{2\times 6}$$

Take the square root of 289.

$$p=\frac{-\left(-7\right)±17}{2\times 6}$$

The opposite of -7 is 7.

$$p=\frac{7±17}{2\times 6}$$

Multiply 2 times 6.

$$p=\frac{7±17}{12}$$

Now solve the equation p=\frac{7±17}{12} when ± is plus. Add 7 to 17.

$$p=\frac{24}{12}$$

Divide 24 by 12.

$$p=2$$

Now solve the equation p=\frac{7±17}{12} when ± is minus. Subtract 17 from 7.

$$p=-\frac{10}{12}$$

Reduce the fraction \frac{-10}{12} to lowest terms by extracting and canceling out 2.

$$p=-\frac{5}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2 for x_{1} and -\frac{5}{6} for x_{2}.

$$6p^{2}-7p-10=6\left(p-2\right)\left(p-\left(-\frac{5}{6}\right)\right)$$

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

$$6p^{2}-7p-10=6\left(p-2\right)\left(p+\frac{5}{6}\right)$$

Add \frac{5}{6} to p by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$6p^{2}-7p-10=6\left(p-2\right)\times \frac{6p+5}{6}$$

Cancel out 6, the greatest common factor in 6 and 6.

$$6p^{2}-7p-10=\left(p-2\right)\left(6p+5\right)$$

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 6

$$x ^ 2 -\frac{7}{6}x -\frac{5}{3} = 0$$

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

$$r + s = \frac{7}{6} rs = -\frac{5}{3}$$

Two numbers r and s sum up to \frac{7}{6} exactly when the average of the two numbers is \frac{1}{2}*\frac{7}{6} = \frac{7}{12}. 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>

$$r = \frac{7}{12} - u s = \frac{7}{12} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{5}{3}

$$(\frac{7}{12} - u) (\frac{7}{12} + u) = -\frac{5}{3}$$

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

$$\frac{49}{144} - u^2 = -\frac{5}{3}$$

Simplify the expression by subtracting \frac{49}{144} on both sides

$$-u^2 = -\frac{5}{3}-\frac{49}{144} = -\frac{289}{144}$$

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

$$u^2 = \frac{289}{144} u = \pm\sqrt{\frac{289}{144}} = \pm \frac{17}{12}$$

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

$$r =\frac{7}{12} - \frac{17}{12} = -0.833 s = \frac{7}{12} + \frac{17}{12} = 2$$

Evaluate

$$\left(p-2\right)\left(6p+5\right)$$