 # Use the discriminant to determine whether each quadratic equation has two unequa Cem Hayes 2021-09-29 Answered
Use the discriminant to determine whether each quadratic equation has two unequal real solutions, a repeated real solution (a double root), or no real solution, without solving the equation.
$$\displaystyle{25}{x}^{{{2}}}-{20}{x}+{4}={0}$$

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Step 1
We know, for a quadratic equation of the form $$\displaystyle{a}{x}^{{{2}}}+{b}{x}+{c}={0},{\left({a}\ne{0}\right)}$$ the discriminant D is given by
$$\displaystyle{D}={b}^{{{2}}}-{4}{a}{c}$$, and depending on the nature of the discriminant the equation the nature of the roots
is identified as follows:
If D>0,Then given quadratic equation has two real and unequal roots.
If D
If D=0,Then given quadratic equation has a repeated real
root(a root with multiplicity 2).
Given equation is: $$\displaystyle{25}{x}^{{{2}}}-{20}{x}+{4}={0},\Rightarrow{a}={25},{b}=-{20},{c}={4}$$.
$$\displaystyle\Rightarrow{D}={b}^{{{2}}}-{4}{a}{c}$$
$$\displaystyle\Rightarrow{D}={\left(-{20}\right)}^{{{2}}}-{4}{\left({25}\right)}{\left({4}\right)}$$
$$\displaystyle\Rightarrow{D}={400}-{400}$$
$$\displaystyle\Rightarrow{D}={0}$$.
Hence given quadratic equation has a repeated real root
(a real root with multiplicity 2).
Step 2
Result:
Given quadratic equation has a single real root with multiplicity 2.