If a and b are roots of the quadratic equation x2 - 4x + 1 = 0, what i

Question

If  a  and  b  are roots of the quadratic equation  x^2 - 4x + 1 = 0 , what is  \sqrt{a}+\sqrt{b} ?

A.  2\sqrt{2}  
 
B.  ±2\sqrt{2}  
 
C.  \sqrt{6} 
 
D.  ±\sqrt{6}  
 
E.  ±8

nick1816 · Answer

a+b = 4; a*b=1

(\sqrt{a}+\sqrt{b})^2

= a+b+2 \sqrt{a*b}

= 4+2*\sqrt{1}

=6

\sqrt{a}+\sqrt{b}  =  \sqrt{6}

MathRevolution · Answer

=>

We have  (x - a)(x - b) = x^2 – (a + b)x + ab = x^2 - 4x + 1.  Thus  a + b = 4  and  ab = 1.

(√a + √b)^2 = (√a + √b)(√a + √b) = a + 2√ab + b = a + b + 2√ab = 4 + 2√1 = 6.

Thus, we have  √a + √b = √6  since  √a  and  √b  are greater than or equal to  0 .

Therefore, C is the answer.
Answer: C

subramanya1991 · Answer

Hello nick,

Can you please mention why you squared both the roots ? I did not understand the logic behind that. Kindly explain.

Thanks.

nick1816 · Answer

We know the value of a+b and a*b from the quadratic equation.

So i transformed the term( \sqrt{a}+\sqrt{b} ), whose value is asked in the question, in terms of a+b and a*b.

MathRevolution · Answer

Standard quadratic equation is  {ax}^2  + bx + c = 0.

On comapring the given equation with standard we get: a = 1, b = -4 and c = 1

Since 'a' and 'b' are the roots of the quadratic equation  x^2 - 4x + 1=0

=> Sum of roots a + b = -  \frac{b}{a}  = -  \frac{(-4) }{ 1}  = 4

=> Product of roots a * b =  \frac{c}{a}  =  \frac{1 }{ 1}  = 1

Squaring (a + b) =>  (\sqrt{a}+\sqrt{b})^2  =  (\sqrt{a})^2 + (\sqrt{b})^2 + 2 \sqrt{a}\sqrt{b}

=>  (\sqrt{a}+\sqrt{b})^2  = a + b + 2  \sqrt{a*b}

=>  (\sqrt{a}+\sqrt{b})^2  = 4 + 2  \sqrt{1}

=>  (\sqrt{a}+\sqrt{b})^2  = 6

=> (a+b) =  \sqrt{6}

Answer C

namya2.0 · Answer

why not +/- root 6 and just + root 6?

Bunuel · Answer

The square root of a number, or more generally, the even root of a number, is always non-negative. Therefore, the sum of the square roots of two numbers (  \sqrt{a}+\sqrt{b} ) cannot be negative.

ManifestDreamMBA · Answer

I think its because we only consider positive root when taking square roots. In GMAT \sqrt{x} results in positive root only whereas for x^2 both positive and negative roots are considered

If a and b are roots of the quadratic equation x2 - 4x + 1 = 0, what i

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If a and b are roots of the quadratic equation x2 - 4x + 1 = 0, what i

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards