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Explain how to tell if a quadratic function has one solution or two solutions.

Question:

Explain how to tell if a quadratic function has one solution or two solutions.

Quadratic Equation:

The number of solutions for a Quadratic equation depend on the Quadratic formula. The history of quadratic formula dates back to late 1500's, when it is believed to have appeared in written texts. It simply says that for a quadratic equation given by {eq}\displaystyle{ax^2+bx+c=0} {/eq}, where {eq}\displaystyle{a,b,c} {/eq} are real constants. The roots of this Quadratic equation can be given by the formula... {eq}\displaystyle{\alpha,\beta=\frac{-b\pm\sqrt{b^2-4ac}}{2a}}\\ {/eq}. This is also called the Quadratic formula.

Answer and Explanation:

We know that the two roots {eq}\alpha,\beta {/eq} of any quadratic equation can be given by the Quadratic formula. But, what if the term under the square root in the Quadratic formula {eq}(b^2-4ac) {/eq} turns out to be {eq}0 {/eq} or {eq}negative {/eq}. We surely know that for most problems that we see in the real world, the square root of a negative number doesn't exist.

So, in cases when the term {eq}\displaystyle{b^2-4ac<0 } {/eq}, there is no solution for such cases.

Now what if the term {eq}\displaystyle{b^2-4ac=0} {/eq}, in such cases there is only one solution to the quadratic equation.

In all other cases, where {eq}\displaystyle{b^2-4ac>0} {/eq}, there are two solutions to the quadratic equation.

Let us see this with some examples for each.

{eq}A. {/eq} No solution

{eq}\displaystyle{x^2+4x+5=0} {/eq}

If we see here, {eq}\displaystyle{a=1,b=4,c=5} {/eq} so we have {eq}\displaystyle{b^2-4ac<0} {/eq}

{eq}B. {/eq} One solution

{eq}\displaystyle{x^2+4x+4=0} {/eq}

If we see here, {eq}\displaystyle{a=1,b=4,c=4} {/eq} so we have {eq}\displaystyle{b^2-4ac=0} {/eq}

{eq}C. {/eq} Two solutions

{eq}\displaystyle{x^2+4x+2=0} {/eq}

If we see here, {eq}\displaystyle{a=1,b=4,c=2} {/eq} so we have {eq}\displaystyle{b^2-4ac>0} {/eq}


Learn more about this topic:

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The Quadratic Formula: Definition & Example

from Remedial Algebra I

Chapter 25 / Lesson 10
11K

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