Solve 2q^2-7q+5

$$2 q ^ { 2 } - 7 q + 5$$

Factor

$$\left(q-1\right)\left(2q-5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 2q^{2}+aq+bq+5. To find a and b, set up a system to be solved.

$$a+b=-7 ab=2\times 5=10$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 10.

$$-1,-10 -2,-5$$

Calculate the sum for each pair.

$$-1-10=-11 -2-5=-7$$

The solution is the pair that gives sum -7.

$$a=-5 b=-2$$

Rewrite 2q^{2}-7q+5 as \left(2q^{2}-5q\right)+\left(-2q+5\right).

$$\left(2q^{2}-5q\right)+\left(-2q+5\right)$$

Factor out q in the first and -1 in the second group.

$$q\left(2q-5\right)-\left(2q-5\right)$$

Factor out common term 2q-5 by using distributive property.

$$\left(2q-5\right)\left(q-1\right)$$

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.

$$2q^{2}-7q+5=0$$

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.

$$q=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 2\times 5}}{2\times 2}$$

Square -7.

$$q=\frac{-\left(-7\right)±\sqrt{49-4\times 2\times 5}}{2\times 2}$$

Multiply -4 times 2.

$$q=\frac{-\left(-7\right)±\sqrt{49-8\times 5}}{2\times 2}$$

Multiply -8 times 5.

$$q=\frac{-\left(-7\right)±\sqrt{49-40}}{2\times 2}$$

Add 49 to -40.

$$q=\frac{-\left(-7\right)±\sqrt{9}}{2\times 2}$$

Take the square root of 9.

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

The opposite of -7 is 7.

$$q=\frac{7±3}{2\times 2}$$

Multiply 2 times 2.

$$q=\frac{7±3}{4}$$

Now solve the equation q=\frac{7±3}{4} when ± is plus. Add 7 to 3.

$$q=\frac{10}{4}$$

Reduce the fraction \frac{10}{4} to lowest terms by extracting and canceling out 2.

$$q=\frac{5}{2}$$

Now solve the equation q=\frac{7±3}{4} when ± is minus. Subtract 3 from 7.

$$q=\frac{4}{4}$$

Divide 4 by 4.

$$q=1$$

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

$$2q^{2}-7q+5=2\left(q-\frac{5}{2}\right)\left(q-1\right)$$

Subtract \frac{5}{2} from q by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$2q^{2}-7q+5=2\times \frac{2q-5}{2}\left(q-1\right)$$

Cancel out 2, the greatest common factor in 2 and 2.

$$2q^{2}-7q+5=\left(2q-5\right)\left(q-1\right)$$

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 2

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

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

$$r + s = \frac{7}{2} rs = \frac{5}{2}$$

Two numbers r and s sum up to \frac{7}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{7}{2} = \frac{7}{4}. 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>

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

To solve for unknown quantity u, substitute these in the product equation rs = \frac{5}{2}

$$(\frac{7}{4} - u) (\frac{7}{4} + u) = \frac{5}{2}$$

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

$$\frac{49}{16} - u^2 = \frac{5}{2}$$

Simplify the expression by subtracting \frac{49}{16} on both sides

$$-u^2 = \frac{5}{2}-\frac{49}{16} = -\frac{9}{16}$$

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

$$u^2 = \frac{9}{16} u = \pm\sqrt{\frac{9}{16}} = \pm \frac{3}{4}$$

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

$$r =\frac{7}{4} - \frac{3}{4} = 1 s = \frac{7}{4} + \frac{3}{4} = 2.500$$

Evaluate

$$\left(q-1\right)\left(2q-5\right)$$