If The Roots Of The Equation $x^2 - 5x - 7 = 0$ Are \[$\alpha, \beta\$\], Find The Equations:(a) Whose Roots Are \[$\alpha^2, \beta^2\$\].(b) Whose Roots Are \[$\alpha + 1, \beta + 1\$\].

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is crucial in various fields such as physics, engineering, and computer science. In this article, we will explore how to find new equations from given roots of a quadratic equation. We will use the quadratic equation $x^2 - 5x - 7 = 0$ and find the equations whose roots are $\alpha^2, \beta^2$ and $\alpha + 1, \beta + 1$.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the roots are given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In our case, the quadratic equation is $x^2 - 5x - 7 = 0$. We can identify the coefficients as $a = 1$, $b = -5$, and $c = -7$.

Finding the Roots

Using the quadratic formula, we can find the roots of the equation:

$$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-7)}}{2(1)}$$

Simplifying the expression, we get:

$$x = \frac{5 \pm \sqrt{25 + 28}}{2}$$

$$x = \frac{5 \pm \sqrt{53}}{2}$$

Therefore, the roots of the equation are $\alpha = \frac{5 + \sqrt{53}}{2}$ and $\beta = \frac{5 - \sqrt{53}}{2}$.

Finding the Equation Whose Roots are $\alpha^2, \beta^2$

To find the equation whose roots are $\alpha^2, \beta^2$, we can use the fact that if $x = \alpha$ is a root of the equation $f(x) = 0$, then $x^2 = \alpha^2$ is a root of the equation $f(x^2) = 0$. Therefore, we can substitute $x^2$ for $x$ in the original equation:

$$(x^2)^2 - 5(x^2) - 7 = 0$$

Expanding the equation, we get:

$$x^4 - 5x^2 - 7 = 0$$

This is the equation whose roots are $\alpha^2, \beta^2$.

Finding the Equation Whose Roots are $\alpha + 1, \beta + 1$

To find the equation whose roots are $\alpha + 1, \beta + 1$, we can use the fact that if $x = \alpha$ is a root of the equation $f(x) = 0$, then $x + 1 = \alpha + 1$ is a root of the equation $f(x + 1) = 0$. Therefore, we can substitute $x + 1$ for $x$ in the original equation:

$$(x + 1)^2 - 5(x + 1) - 7 = 0$$

Expanding the equation, we get:

$$x^2 + 2x + 1 - 5x - 5 - 7 = 0$$

Simplifying the equation, we get:

$$x^2 - 3x - 11 = 0$$

This is the equation whose roots are $\alpha + 1, \beta + 1$.

Conclusion

In this article, we have shown how to find new equations from given roots of a quadratic equation. We have used the quadratic equation $x^2 - 5x - 7 = 0$ and found the equations whose roots are $\alpha^2, \beta^2$ and $\alpha + 1, \beta + 1$. We have used the quadratic formula to find the roots of the equation and then used substitution to find the new equations. This technique can be used to find new equations from given roots of any quadratic equation.

References

• [1] "Quadratic Formula" by Math Open Reference. Retrieved February 2023.
• [2] "Quadratic Equations" by Khan Academy. Retrieved February 2023.

Further Reading

• [1] "Algebra" by Michael Artin. Retrieved February 2023.
• [2] "Calculus" by Michael Spivak. Retrieved February 2023.

Glossary

• Quadratic Equation: An equation of the form $ax^2 + bx + c = 0$.
• Roots: The solutions to a quadratic equation.
• Quadratic Formula: A formula for finding the roots of a quadratic equation.
• Substitution: A technique for finding new equations from given roots of a quadratic equation.
  Quadratic Equations: A Q&A Guide =====================================

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is crucial in various fields such as physics, engineering, and computer science. In this article, we will provide a Q&A guide to help you understand quadratic equations and how to solve them.

Q: What is a quadratic equation?

A: A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including:

• Factoring: If the equation can be factored into the product of two binomials, you can solve it by setting each factor equal to zero.
• Quadratic formula: The quadratic formula is a formula for finding the roots of a quadratic equation. It is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Graphing: You can graph the quadratic equation and find the roots by finding the points where the graph intersects the x-axis.

Q: What is the quadratic formula?

A: The quadratic formula is a formula for finding the roots of a quadratic equation. It is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the coefficients $a$, $b$, and $c$ in the quadratic equation. Then, you can plug these values into the formula and simplify to find the roots.

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula is a general method for solving quadratic equations, while factoring is a specific method that works only when the equation can be factored into the product of two binomials.

Q: Can I use the quadratic formula to solve any quadratic equation?

A: Yes, the quadratic formula can be used to solve any quadratic equation, regardless of whether it can be factored or not.

Q: How do I find the roots of a quadratic equation using the quadratic formula?

A: To find the roots of a quadratic equation using the quadratic formula, you need to:

1. Identify the coefficients $a$, $b$, and $c$ in the quadratic equation.
2. Plug these values into the quadratic formula.
3. Simplify the expression to find the roots.

Q: What is the significance of the discriminant in the quadratic formula?

A: The discriminant is the expression under the square root in the quadratic formula, given by $b^2 - 4ac$. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has no real roots.

Q: Can I use the quadratic formula to solve quadratic equations with complex roots?

A: Yes, the quadratic formula can be used to solve quadratic equations with complex roots. In this case, the roots will be complex numbers.

Q: How do I find the equation whose roots are $\alpha^2, \beta^2$?

A: To find the equation whose roots are $\alpha^2, \beta^2$, you can use the fact that if $x = \alpha$ is a root of the equation $f(x) = 0$, then $x^2 = \alpha^2$ is a root of the equation $f(x^2) = 0$. Therefore, you can substitute $x^2$ for $x$ in the original equation and simplify to find the new equation.

Q: How do I find the equation whose roots are $\alpha + 1, \beta + 1$?

A: To find the equation whose roots are $\alpha + 1, \beta + 1$, you can use the fact that if $x = \alpha$ is a root of the equation $f(x) = 0$, then $x + 1 = \alpha + 1$ is a root of the equation $f(x + 1) = 0$. Therefore, you can substitute $x + 1$ for $x$ in the original equation and simplify to find the new equation.

Conclusion

In this article, we have provided a Q&A guide to help you understand quadratic equations and how to solve them. We have covered topics such as the quadratic formula, factoring, graphing, and finding new equations from given roots. We hope that this guide has been helpful in your understanding of quadratic equations.