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This is the second of our many quizzes on the zero product principle. Click on this to access more such competitive quizzes on polynomials.

Zero Product Principle Quiz 2

mathematics

 Solve for $x: 15{{x}^{2}}-2{{x}^{3}}=4x+21$

 Shift all the terms on the LHS and then convert the polynomial in factored form. Now put each factor equal to zero to find the required answer.

  $\begin{align} & 15{{x}^{2}}-2{{x}^{3}}=4x+21 \\ & \Rightarrow 2{{x}^{3}}-15{{x}^{2}}+4x+21=0 \\ & \text{By putting x = -1, we get LHS = RHS = 0.}\\ & \text{So, x + 1 is a factor of the above polynomial.}\\ & \Rightarrow (x+1)(2{{x}^{2}}-17x+21)=0 \\ & \Rightarrow (x+1)(2{{x}^{2}}-14x-3x+21)=0 \\ & \Rightarrow (x+1)(2x-3)(x-7)=0 \\ & \Rightarrow x=-1,7,\dfrac{3}{2} \end{align}$ 

 If $s = 9x - 5$ and $t = 27 – 39x$, then for what values of $x$ is ${{s}^{2}}=t$? 

 Shift $t$ on the LHS and then factorize the quadratic equation. Now put each factor equal to zero to find the required answer. 

  $\begin{align} & {{s}^{2}}=t\Rightarrow {{s}^{2}}-t=0 \\ & \Rightarrow {{(9x-5)}^{2}}+39x-27=0 \\ & \Rightarrow 81{{x}^{2}}-51x-2=0 \\ & \Rightarrow 81{{x}^{2}}-54x+3x-2=0 \\ & \Rightarrow (27x+1)(3x-2)=0 \\ & \Rightarrow x=\dfrac{2}{3},\dfrac{-1}{27} \end{align}$ 

 If a first number is 5 more than the thrice of the second number and the sum of the squares of both the numbers is 305, then find the first number $(\text{Given}: x > 0)$. 

 Assume second number be $x$ and accordingly simplify first number. Apply the given condition and convert it into factored form to solve for $x$. 

 $\begin{align} & \text{Let the second number be “x”. } \\ & \text{Then, the first number }= 3x + 5.\\ & {{x}^{2}}+{{(3x+5)}^{2}}=305 \\ & \Rightarrow 10{{x}^{2}}+30x-280=0 \\ & \Rightarrow {{x}^{2}}+7x-4x-28=0 \\ & \Rightarrow x=-7,4 \\ & \therefore \text{First number}=3x+5=17 \end{align}$

 George is 5 years younger than Riya and Krish’s age is the product of the ages of George and Riya. If the sum of the age of George, Riya and Krish is 83, then find the age of Krish?

 Take George’s age as $x$ and accordingly solve for the age of Riya and Krish. Now, apply the given condition and put each factor of equation equal to zero to solve for $x$.

 $\begin{align} & \text{Let George’s age be “x” years.} \\ & \text{Then, Riya’s age is x + 5 years and Krish’s age x(x + 5) years.}\\ & \Rightarrow x+x+5+{{x}^{2}}+5x=83 \\ & \Rightarrow {{x}^{2}}+7x-78=0 \\ & \Rightarrow {{x}^{2}}+13x-6x-78=0 \\ & \Rightarrow (x+13)(x-6)=0 \\ & \Rightarrow \text{George’s age = x = 6years }\\ & \therefore \text{Krish’s age = 6(6 + 5) = 66years} \end{align}$

 If G, H and I are the three points of triangle GHI whose area is given by $5{{x}^{2}}-17x+14=0$, then for what possible values of $x$ points G, H and I will be collinear?

 Put Area of $\Delta \text{GHI}=0$ and then factorize the equation to get the required answer.

 $\begin{align} & \text{Points G, H and I will be collinear if Area of }\Delta \text{GHI}=0\\ & \Rightarrow 5{{x}^{2}}-17x+14=0 \\ & \Rightarrow 5{{x}^{2}}-10x-7x+14=0 \\ & \Rightarrow (5x-7)(x-2)=0 \\ & \Rightarrow x=\dfrac{7}{5},2 \end{align}$ 