Question

Find the values of p for which the following quadratic equation has two equal roots:(p-12)x^2 - 2(p-12)x + 2 = 0

Answer 1

Answer :

The value of p is either 14 or 12

Given :

The quadratic equation :

• (p -12)x² - 2(p - 12)x + 2 = 0
• It has two equal roots .

To Find :

• The value of p

Solution :

We know that if a quadratic equation has two equal roots , its discriminant , b² - 4ac = 0 .

Here , in the equation

(p - 12)x² - 2(p - 12)x + 2 =0

a = (p - 12) , b = -2(p - 12) and c = 2

Therefore ,

$\sf \implies \{ - 2 (p - 12) \}^{2} - 4\{ (p - 12) \times 2 \} = 0 \\\\ \sf \implies 4 \{ (p - 12)^{2} - 2(p - 12) \} = 0 \\\\ \sf \implies p^{2} - 24p + 144 - 2p + 24 = 0 \\\\ \sf \implies p^{2} - 26p + 168 = 0 \\\\ \sf \implies p^{2} - 12p - 14p + 168 = 0 \\\\ \sf \implies p(p - 12) - 14(p - 12) = 0 \\\\ \sf \implies (p - 12)(p - 14) = 0$

Thus we have

$\sf \implies p - 12 =0 \: \: and \: \: p - 14 = 0 \\\\ \sf \implies p = 12 \: \: and \: \implies p = 14$

Concept Used here :

Discriminant of a quadratic equation is given by → b² - 4ac , where :

• a is the coefficient of x²

• b is the coefficient of x

• c is the constant term

Nature of the roots according to discriminant :

→ If discriminant , b² - 4ac ≥ 0 then the roots are real

→ If discriminant , b² - 4ac = 0 , then the roots are equal and real

→ If discriminant , b² - 4ac < 0 , then their exists no real roots ( i.e. immaginary roots are obtained)

Answer 2

$\bigstar$ Answer :

$\star$ The value of p is 14 or 12

$\bigstar$ Given :

$\star$ Given that , The quadratic equation is  (p -12)x² - 2(p - 12)x + 2 = 0

and It has two equal roots .

$\bigstar$ To Find :

$\star$ The value of p in quadratic equation , (p -12)x² - 2(p - 12)x + 2 = 0

$\bigstar$ Solution :

$\star$ We know that if a quadratic equation ax²+bx+c , has two equal roots ,

its discriminant , b² - 4ac = 0 .

$\star$ Now compare the given equation  (p - 12)x² - 2(p - 12)x + 2 =0 , with

ax²+bx+c , we get ,

$\star$ a = (p - 12) , b = -2(p - 12) and c = 2

$\star$ Therefore ,

$=>b^2-4ac=0\\\\=>[(-2(p-12))^2-4(p-12)(2]=0\\\\=>[4(p-12)^2-8(p-12)]=0\\\\=>4[(p-12)^2-2(p-12)]=0\\\\=>[p^2+144-24p-2p+24]=0\\\\=>[p^2-26p+168]=0\\\\=>p^2-14p-12p+168=0\\\\=>p(p-14)-12(p-14)=0\\\\=>(p-14)(p-12)=0$

$\star$ Therefore ,

=> p - 14 = 0 and p - 12 = 0

=> p = 14 and p = 12

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$\bigstar$ Concept related to quadratic equation :

$\star$ Discriminant of a quadratic equation ax²+bx+c  is given by → b² - 4ac ,  where :

• a is the coefficient of x²
• b is the coefficient of x
• c is the constant term

$\star$ Nature of the roots according to discriminant :

• If discriminant , b² - 4ac ≥ 0 then the roots are real
• If discriminant , b² - 4ac = 0 , then the roots are equal and real
• If discriminant , b² - 4ac < 0 , then their exists no real roots ( i.e. imaginary roots are obtained)

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