SOLUTION: When the parabola y=x^2-a(x+1)+3 intersects the x-axis at one point then a= ? Or ?

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Question 879623: When the parabola y=x^2-a(x+1)+3 intersects the x-axis at one point then a= ? Or ?
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
y=x%5E2-a%28x%2B1%29%2B3
y=x%5E2-ax-a%2B3, you should c.... you can use the general solution to a quadratic equation. Watch for defined variables in the present equation and not confuse them with the instructionally typical variables.

y=x%5E2-ax%2B%283-a%29

Fitting the parts into the formula places,
roots are x=%28a%2B-+sqrt%28a%5E2-4%283-a%29%29%29%2F2
x=%28a%2B-+sqrt%28a%5E2-12%2B4a%29%29%2F2

Because you want x to be only a single value as a root, you want the discriminant, a%5E2%2B4a-12=0. FACTORABLE:
%28a-2%29%28a%2B6%29=0
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a=2 OR a=-6
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CONCEPTUAL DISCUSSION

"y intersects the x axis", means y=0, so that the solutions for x are the "roots" of y.

When we solve this for the equation that you were given, y=x%5E2-a%28x%2B1%29%2B3=0, the application of the general solution of a quadratic formula gives us that x is this formula or expression:

highlight_green%28x=%28a%2B-+sqrt%28a%5E2-12%2B4a%29%29%2F2%29.
This or these are the x values for which y crosses or touches the x-axis.
If you specify that y has exactly ONE root, then y intersects the x-axis at ONE point only. This can occur only for sqrt%28a%5E2-12%2Ba%29=0, from the solution of x. The expression a%5E2-12%2Ba from your solutions of x is the discriminant.

The typical model for instruction in discussing a quadratic formula is much expressed as Ax%5E2%2BBx%2BC=0, but typically the lower-case coefficients and constant are used. The discriminant for this general quadratic equation would be B%5E2-4%2AA%2AC. In keeping with the known and derivable generality about roots of the equation, the discriminant is zero if there is only one root (or only one x-intercept.)