Solve -7x^2+48x+7

$$- 7 x ^ { 2 } + 48 x + 7$$

Factor

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

Solution Steps

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

$$a+b=48 ab=-7\times 7=-49$$

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 -49.

$$-1,49 -7,7$$

Calculate the sum for each pair.

$$-1+49=48 -7+7=0$$

The solution is the pair that gives sum 48.

$$a=49 b=-1$$

Rewrite -7x^{2}+48x+7 as \left(-7x^{2}+49x\right)+\left(-x+7\right).

$$\left(-7x^{2}+49x\right)+\left(-x+7\right)$$

Factor out 7x in -7x^{2}+49x.

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

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

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

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.

$$-7x^{2}+48x+7=0$$

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.

$$x=\frac{-48±\sqrt{48^{2}-4\left(-7\right)\times 7}}{2\left(-7\right)}$$

Square 48.

$$x=\frac{-48±\sqrt{2304-4\left(-7\right)\times 7}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-48±\sqrt{2304+28\times 7}}{2\left(-7\right)}$$

Multiply 28 times 7.

$$x=\frac{-48±\sqrt{2304+196}}{2\left(-7\right)}$$

Add 2304 to 196.

$$x=\frac{-48±\sqrt{2500}}{2\left(-7\right)}$$

Take the square root of 2500.

$$x=\frac{-48±50}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-48±50}{-14}$$

Now solve the equation x=\frac{-48±50}{-14} when ± is plus. Add -48 to 50.

$$x=\frac{2}{-14}$$

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

$$x=-\frac{1}{7}$$

Now solve the equation x=\frac{-48±50}{-14} when ± is minus. Subtract 50 from -48.

$$x=-\frac{98}{-14}$$

Divide -98 by -14.

$$x=7$$

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

$$-7x^{2}+48x+7=-7\left(x-\left(-\frac{1}{7}\right)\right)\left(x-7\right)$$

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

$$-7x^{2}+48x+7=-7\left(x+\frac{1}{7}\right)\left(x-7\right)$$

Add \frac{1}{7} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-7x^{2}+48x+7=-7\times \frac{-7x-1}{-7}\left(x-7\right)$$

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

$$-7x^{2}+48x+7=\left(-7x-1\right)\left(x-7\right)$$

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.

$$x ^ 2 -\frac{48}{7}x -1 = 0$$

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

$$r + s = \frac{48}{7} rs = -1$$

Two numbers r and s sum up to \frac{48}{7} exactly when the average of the two numbers is \frac{1}{2}*\frac{48}{7} = \frac{24}{7}. 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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{24}{7} - u s = \frac{24}{7} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -1

$$(\frac{24}{7} - u) (\frac{24}{7} + u) = -1$$

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

$$\frac{576}{49} - u^2 = -1$$

Simplify the expression by subtracting \frac{576}{49} on both sides

$$-u^2 = -1-\frac{576}{49} = -\frac{625}{49}$$

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

$$u^2 = \frac{625}{49} u = \pm\sqrt{\frac{625}{49}} = \pm \frac{25}{7}$$

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

$$r =\frac{24}{7} - \frac{25}{7} = -0.143 s = \frac{24}{7} + \frac{25}{7} = 7$$

Evaluate

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