How to Find x given y of Quadratic Functions

How to find x given y of quadratic equations

Welcome to the intriguing world of quadratic functions, where we delve into the art of finding \(x\) when you’re armed with \(y\). If you’ve ever wondered how to unlock the mysteries of quadratic equations and trace their solutions, you’re about to embark on an enlightening journey. In this article, we’ll guide you through the step-by-step process of solving for \(x\) in quadratic functions. Whether you’re a math enthusiast, a student facing algebraic challenges, or simply curious about quadratic equations, this guide will equip you with the skills to unveil the value of \(x\) with confidence and precision. Let’s dive into the world of quadratic functions and make math magic happen!

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Question 1

Find any values of \( x \) for \( y = x^2+x \) if \( y=2 \).

\( \begin{align} x^2+x &= 2 \\ x^2+x-2 &= 0 \\ (x-1)(x+2) &= 0 \\ \therefore x &=1 \text{ or } x=-2 \end{align} \)

Question 2

Find any values of \( x \) for \( y = x^2-6x+10 \) if \( y=1 \).

\( \begin{align} x^2-6x+10 &= 1 \\ x^2-6x+9 &= 0 \\ (x-3)^2 &= 0 \\ \therefore x=3 \end{align} \)

Question 3

Find any values of \( x \) for \( y = 3x^2 \) if \( y=12 \).

\( \begin{align} 3x^2 &= 12 \\ x^2 &= 4 \\ x &= \pm \sqrt{4} \\ \therefore x &= \pm 2 \end{align} \)

Question 4

Find any values of \( x \) for \( y = x^2+2x \) if \( y=1 \).

\( \begin{align} \displaystyle x^2+2x &= 1 \\ x^2+2x-1 &= 0 \\ x &= \frac{-2 \pm \sqrt{2^2-4 \times 1 \times -1}}{2 \times 1} \\ &= \frac{-2 \pm \sqrt{8}}{2} \\ &= \frac{-2 \pm 2\sqrt{2}}{2} \\ \therefore x &= -1 \pm \sqrt{2} \end{align} \)

Question 5

Find any values of \( x \) for \( y = x^2+2x+3 \) if \( y=1 \).

\( \begin{align} \displaystyle x^2+2x+3 &= 1 \\ x^2+2x+2 &= 0 \\ x &= \frac{-2 \pm \sqrt{2^2-4 \times 1 \times 2}}{2 \times 1} \\ &= \frac{-2 \pm \sqrt{-4}}{2} \\ \therefore &\text{No real values for } x \end{align} \)

Conclusion

Finding \(x\) given \(y\) in quadratic functions may seem daunting at first, but by following these steps and using the quadratic formula, you can confidently navigate the world of quadratic equations. Remember, practice makes perfect. As you work through more examples and gain experience, you’ll become proficient in solving for \(x\) in various quadratic scenarios. So, embrace the challenge, keep those equations coming, and let the magic of math unfold before your eyes.

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