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The number of real solutions of the equation \( \Large |x^{2}+4x+3|+2x+5=0 \) are:

A) 1
B) 2
C) 3
D) 4

Correct Answer:


B) 2

Description for Correct answer:

We have \( \Large |x^{2}+4x+3|+2x+5=0 \)

have two cases arise

Case l: When \( \Large x^{2}+4x+3+2x+5=0 \)

=> \( \Large x^{2}+6x+8=0 \)

=> \( \Large \left(x+2\right) \left(x+4\right)=0 \)

=> \( \Large x = -2, -4 \)

x = -2 is not satisfying the condition \( \Large x^{2}+4x+3>0 \),

so. x = -4 is the only solution of given equation.

Case II: When \( \Large x^{2}+4x+3<0 \)

This gives \( \Large - \left(x^{2}+4x+3\right)+2x+5=0 \)

=> \( \Large -x^{2}-2x+2=0 \)

=> \( \Large x^{2}+2x-2=0 \)

=> \( \Large \left(x+1+\sqrt{3}\right) \left(x+1-\sqrt{3}\right)=0 \)

=> \( \Large x = -1+\sqrt{3}, -1-\sqrt{3} \)

Hence, \( \Large x=- \left(1+\sqrt{3}\right) \) satisfy the given condition.

Since, \( \Large x^{2}+4x+3<0 \) while \( \Large x= -1+\sqrt{3} \) is not satisfying the condition. Thus number of real solutions are two.

Part of solved Quadratic Equations questions and answers : >> Elementary Mathematics >> Quadratic Equations

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Similar Questions

1). If the roots of the given equation:
\( \Large \left(\cos p-1\right)x^{2}+ \left(\cos p\right)x+\sin p = 0 \) are real, then:

A). \( \Large P \epsilon \left(- \pi ,0\right) \)

B). \( \Large P \epsilon \left(- \frac{ \pi }{2}, \frac{ \pi }{2} \right) \)

C). \( \Large P \epsilon \left(0, \pi \right) \)

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A). \( \Large \pm \frac{1}{9}, \pm 2 \)

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3). The number of solution of \( \Large \frac{log 5 + log \left(x^{2}+1\right) }{log \left(x-2\right) }=2 \)

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4). If the expression \( \Large \left(mx-1+\frac{1}{x}\right) \) is always nonnegative, then the minimum value of m must be:

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5). The value of x in the given equation \( \Large 4^{x}-3^{x-\frac{1}{2}}=3^{x+\frac{1}{2}}-2^{2x-1} \) is:

A). \( \Large \frac{4}{3} \)

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6). The harmonic mean of the roots of equation \( \Large \left(5+\sqrt{2}x^{2}-14+\sqrt{5}\right)x+8+2\sqrt{5}=0 \) is:

A). 2

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7). For what value of \( \Large \lambda \) the sum of the squares of the roots of \( \Large x^{2}+ \left(2+\lambda\right)n-\frac{1}{2} \left(1+\lambda\right)=0 \) is minimum?

A). \( \Large \frac{3}{2} \)

B). 1

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D). \( \Large \frac{11}{4} \)

8). Given that \( \Large \tan \alpha \) and \( \Large \tan \beta \) are the roots of \( \Large x^{2}-px+q=0 \), then the value of \( \Large \sin^{2} \left(\alpha + \beta \right) \) is equal to:

A). \( \Large \frac{p^{2}}{p^{2}+ \left(1-q\right)^{2} } \)

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D). \( \Large \frac{p^{2}}{ \left(p+q\right)^{2} } \)

9). The number of values of K for which the system of equations \( \Large \left(K+1\right)x+84=4K \) and \( \Large Kx+ \left(K+3\right)y=3 K-1 \) has infinitely many solution, is:

A). 0

B). 1

C). 2

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10). The set of all real numbers x for which \( \Large x^{2}-|x+2|+x>0 \) is:

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B). \( \Large \left(-\infty, -\sqrt{2}\right) \ \left(\sqrt{2}, \infty\right) \)

C). \( \Large \left(-\infty, -1\right) \ \left(1, \infty\right) \)

D). \( \Large \left(\sqrt{2}, \infty\right) \)