What is 5x^2-28x+19=0?

2 Answers
Feb 21, 2018

See explanation below!

Explanation:

Recall that a linear equation in one variable is of the form ax+b=0, where a and b are constants and a≠0.

For example: " " 3x+5=0

A quadratic equation has an x^2 (x-squared) term. ("Quadratum" is Latin for square.) The general quadratic equation in standard form looks like:

ax^2+bx+c=0 \ \ \ \cdots\ \ \cdots where a\ne 0

If we want to find the x or x's that work, we might guess and substitute and hope we get lucky, or we might try one of these four methods:

We can solve graphically by equating the polynomial to y instead of to 0, we get an equation whose graph is a parabola. The x-\text{intercepts} of the parabola (if any) correspond to the solutions of the original quadratic equation.

Feb 21, 2018

The solutions are x=(14+-sqrt101)/5.

Explanation:

One way to find the solutions to a quadratic is to use the quadratic formula:

x=(-b+-sqrt(b^2-4ac))/(2a)

Here's our quadratic:

5x^2-28x+19=0

The values are a=5, b=-28, and c=19. Plug in the values to the equation:

x=(-b+-sqrt(b^2-4ac))/(2a)

color(white)x=(-(-28)+-sqrt((-28)^2-4(5)(19)))/(2(5))

color(white)x=(28+-sqrt((-28)^2-4(5)(19)))/10

color(white)x=(28+-sqrt(784-4(5)(19)))/10

color(white)x=(28+-sqrt(784-380))/10

color(white)x=(28+-sqrt(404))/10

color(white)x=(28+-sqrt(4*101))/10

color(white)x=(28+-sqrt(2^2*101))/10

color(white)x=(28+-2sqrt(101))/10

color(white)x=(14+-sqrt(101))/5

This is as simplified as the answer gets. The two final solutions are:

x=(14+sqrt101)/5
and
x=(14-sqrt101)/5

Here's the graph of the function (with an altered scale):

graph{5x^2-28x+19[-3,8,-30,20]}