Solve f(x)=7x^2-3x-10

$$f ( x ) = 7 x ^ { 2 } - 3 x - 10$$

Factor

$$\left(7x-10\right)\left(x+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 7x^{2}+ax+bx-10. To find a and b, set up a system to be solved.

$$a+b=-3 ab=7\left(-10\right)=-70$$

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 -70.

$$1,-70 2,-35 5,-14 7,-10$$

Calculate the sum for each pair.

$$1-70=-69 2-35=-33 5-14=-9 7-10=-3$$

The solution is the pair that gives sum -3.

$$a=-10 b=7$$

Rewrite 7x^{2}-3x-10 as \left(7x^{2}-10x\right)+\left(7x-10\right).

$$\left(7x^{2}-10x\right)+\left(7x-10\right)$$

Factor out x in 7x^{2}-10x.

$$x\left(7x-10\right)+7x-10$$

Factor out common term 7x-10 by using distributive property.

$$\left(7x-10\right)\left(x+1\right)$$

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.

$$7x^{2}-3x-10=0$$

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.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 7\left(-10\right)}}{2\times 7}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 7\left(-10\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-3\right)±\sqrt{9-28\left(-10\right)}}{2\times 7}$$

Multiply -28 times -10.

$$x=\frac{-\left(-3\right)±\sqrt{9+280}}{2\times 7}$$

Add 9 to 280.

$$x=\frac{-\left(-3\right)±\sqrt{289}}{2\times 7}$$

Take the square root of 289.

$$x=\frac{-\left(-3\right)±17}{2\times 7}$$

The opposite of -3 is 3.

$$x=\frac{3±17}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{3±17}{14}$$

Now solve the equation x=\frac{3±17}{14} when ± is plus. Add 3 to 17.

$$x=\frac{20}{14}$$

Reduce the fraction \frac{20}{14} to lowest terms by extracting and canceling out 2.

$$x=\frac{10}{7}$$

Now solve the equation x=\frac{3±17}{14} when ± is minus. Subtract 17 from 3.

$$x=-\frac{14}{14}$$

Divide -14 by 14.

$$x=-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{10}{7} for x_{1} and -1 for x_{2}.

$$7x^{2}-3x-10=7\left(x-\frac{10}{7}\right)\left(x-\left(-1\right)\right)$$

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

$$7x^{2}-3x-10=7\left(x-\frac{10}{7}\right)\left(x+1\right)$$

Subtract \frac{10}{7} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$7x^{2}-3x-10=7\times \frac{7x-10}{7}\left(x+1\right)$$

Cancel out 7, the greatest common factor in 7 and 7.

$$7x^{2}-3x-10=\left(7x-10\right)\left(x+1\right)$$

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 7

$$x ^ 2 -\frac{3}{7}x -\frac{10}{7} = 0$$

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

$$r + s = \frac{3}{7} rs = -\frac{10}{7}$$

Two numbers r and s sum up to \frac{3}{7} exactly when the average of the two numbers is \frac{1}{2}*\frac{3}{7} = \frac{3}{14}. 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>

$$r = \frac{3}{14} - u s = \frac{3}{14} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{10}{7}

$$(\frac{3}{14} - u) (\frac{3}{14} + u) = -\frac{10}{7}$$

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

$$\frac{9}{196} - u^2 = -\frac{10}{7}$$

Simplify the expression by subtracting \frac{9}{196} on both sides

$$-u^2 = -\frac{10}{7}-\frac{9}{196} = -\frac{289}{196}$$

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

$$u^2 = \frac{289}{196} u = \pm\sqrt{\frac{289}{196}} = \pm \frac{17}{14}$$

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

$$r =\frac{3}{14} - \frac{17}{14} = -1.000 s = \frac{3}{14} + \frac{17}{14} = 1.429$$

Evaluate

$$\left(7x-10\right)\left(x+1\right)$$