Use factoring to�find the x intercepts of the graphs of these quadratic poly-nomials:y=ax2+bx+c,where a,b,c are constants
the quadratic polynomial is y = ax^2+bx+c
a x^2 + b x = 0 - c
x^2 + (b/a) x + b^2 / 4a^2 = - c/a + b^2/ 4a^2
now factor
(x + b / 2a )^2 = - 4ac / 4 a^2 + b^2 / 4a^2
x + b/2a = +/- sqrt [ (b^2-4ac) / 4a^2 ]
x = -b/2a +/- sqrt(b^2-4ac) /2a
x = [ -b +/- sqrt (b^2 - 4 ac ) ] / 2a
look familiar?
To find the x-intercepts of a quadratic polynomial, you can use factoring. The x-intercepts represent the points where the graph crosses or intersects the x-axis. In other words, they are the values of x for which the y-coordinate is zero.
Here's the step-by-step process to find the x-intercepts using factoring:
Step 1: Start with a quadratic polynomial in the form of y = ax^2 + bx + c.
Step 2: Set y equal to zero since we are looking for the x-intercepts. So, the equation becomes 0 = ax^2 + bx + c.
Step 3: Factor the quadratic equation if possible. This involves finding the two binomials that multiply together to give you the quadratic equation.
Step 4: Set each factor equal to zero and solve for x. This step gives you the values of x at which the polynomial intersects the x-axis.
Step 5: The solutions you find in step 4 are the x-intercepts of the graph.
For example, let's say we have the quadratic polynomial y = 2x^2 + 5x - 3, and we want to find the x-intercepts.
Step 1: Start with y = 2x^2 + 5x - 3.
Step 2: Set y equal to zero: 0 = 2x^2 + 5x - 3.
Step 3: Factor the quadratic equation: 0 = (2x - 1)(x + 3).
Step 4: Set each factor equal to zero:
2x - 1 = 0 and x + 3 = 0.
Solving these equations, we find:
x = 1/2 and x = -3.
Step 5: The x-intercepts are x = 1/2 and x = -3.
So, the x-intercepts of the graph of the quadratic polynomial y = 2x^2 + 5x - 3 are x = 1/2 and x = -3.