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Author Jonathan David Author of Fiction
Author Jonathan David
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Radical Equations Quadratic in Form | Solve x² − 3x − √(x² − 3x) = 2

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Sample from The Ultimate Crash Course Series
Get the full bundle with structured lessons and over 1,000 lessons and podcasts via Payhip and theSTEMmajor.com.

Radical Equations Quadratic in Form

Find the real solutions of

\displaystyle x^2-3x-\sqrt{x^2-3x}=2

Step 1: Recognize the Quadratic Form

This radical equation becomes much easier if we treat the repeated expression as a single variable. Let

\displaystyle u=\sqrt{x^2-3x}

Then

\displaystyle u^2=x^2-3x

Substitute these into the original equation:

\displaystyle u^2-u=2

Step 2: Move Everything to One Side

\displaystyle u^2-u-2=0

Step 3: Factor the Quadratic

\displaystyle (u-2)(u+1)=0

So the possible values are

\displaystyle u=2 \quad \text{or} \quad u=-1

Step 4: Use the Radical Restriction

Because

\displaystyle u=\sqrt{x^2-3x}

a square root must be nonnegative. Therefore

\displaystyle u=-1

is not valid, so we keep only

\displaystyle u=2

Step 5: Substitute Back

\displaystyle \sqrt{x^2-3x}=2

Square both sides:

\displaystyle x^2-3x=4

\displaystyle x^2-3x-4=0

Step 6: Factor Again

\displaystyle (x-4)(x+1)=0

\displaystyle x=4 \quad \text{or} \quad x=-1

Step 7: Check the Solutions

Check x=4:

\displaystyle 4^2-3(4)-\sqrt{4^2-3(4)}=16-12-\sqrt{16-12}=4-2=2

Check x=-1:

\displaystyle (-1)^2-3(-1)-\sqrt{(-1)^2-3(-1)}=1+3-\sqrt{1+3}=4-2=2

Both values work.

Final Answer

\displaystyle x=-1 \quad \text{or} \quad x=4

Why This Method Works

This is a radical equation that is quadratic in form. Instead of expanding into a quartic expression, the cleaner method is substitution. When the same algebraic expression appears both inside a square root and outside it, replacing that repeating structure with a temporary variable often turns the problem into an ordinary quadratic.

Sample from The Ultimate Crash Course Series
This lesson is part of the Ultimate Crash Course bundle. Get the full collection at Payhip and access over 1,000 lessons and podcasts through theSTEMmajor.com. 





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