Learn how to solve for what values of x is x^2-36=5x true and understand the different techniques used to solve quadratic equations. Find out more here.

Introduction

The equation x^2-36=5x is a quadratic equation that has puzzled many students for years. It is a simple equation that requires a bit of algebraic manipulation to solve. In this article, we will explore the different techniques used to solve this equation and determine for what values of x it is true.

Understanding the Equation x^2-36=5x

The equation x^2-36=5x is a basic quadratic equation. It is in the form ax^2+bx+c=0, where a=1, b=-5, and c=-36. The first step in solving this equation is to move all the terms to one side of the equation. In this case, we can subtract 5x from both sides of the equation, which gives us x^2-5x-36=0.

Now that the equation is in standard form, we need to find the roots of the equation. The roots of a quadratic equation are the values of x that make the equation equal to zero. There are different techniques used to find the roots of a quadratic equation, including factoring, completing the square, and using the quadratic formula. In this case, we will use the quadratic formula to find the roots of the equation.

Solving for the Roots of the Equation

The quadratic formula is a formula used to find the roots of a quadratic equation. The formula is given as follows: x = (-b ± sqrt(b^2-4ac)) / 2a. In this case, a=1, b=-5, and c=-36. Substituting these values into the formula, we get x = (5 ± sqrt(5^2-4(1)(-36))) / 2(1).

Simplifying the equation, we get x = (5 ± sqrt(169)) / 2. Taking the square root of 169, we get x = (5 ± 13) / 2. Therefore, the roots of the equation are x = 9 and x = -4.

Using the Discriminant to Determine the Nature of the Roots

The discriminant is a value used to determine the nature of the roots of a quadratic equation. The discriminant is given by b^2-4ac. If the discriminant is greater than zero, the equation has two real roots. If the discriminant is equal to zero, the equation has one real root. If the discriminant is less than zero, the equation has two complex roots.

In this case, b^2-4ac = (-5)^2-4(1)(-36) = 169. Since the discriminant is greater than zero, the equation has two real roots.

Graphical Representation of the Equation

It is always helpful to represent an equation graphically. We can use a graphing calculator or software to graph the equation y=x^2-5x-36. The graph of the equation is a parabola that opens upward and intersects the x-axis at x = 9 and x = -4.

Conclusion

In conclusion, we have explored the different techniques used to solve the equation x^2-36=5x. We have found that the equation has two real roots, x = 9 and x = -4. Graphically, the equation is represented by a parabola that opens upward and intersects the x-axis at x = 9 and x = -4.

Solving for x using Factoring

Another way to find the roots of a quadratic equation is by factoring. In this case, we can factor the equation x^2-5x-36=0 into (x-9)(x+4)=0. From this, we can see that x-9=0 or x+4=0. Therefore, the roots of the equation are x=9 and x=-4.

Using factoring to solve quadratic equations is a useful technique, especially for equations that can be factored easily. However, not all quadratic equations can be factored, and in those cases, we have to use other techniques like completing the square or the quadratic formula.

Using Completing the Square to Find the Roots

Completing the square is another technique used to find the roots of a quadratic equation. The first step is to write the equation in the form x^2+bx=c. In this case, we can add 36 to both sides of the equation to get x^2-5x=36. To complete the square, we need to add (b/2)^2 to both sides of the equation. In this case, b=-5, so (b/2)^2 = (-5/2)^2 = 6.25. Adding 6.25 to both sides of the equation, we get x^2-5x+6.25=42.25.

Now that the equation is in the form (x-b)^2=c, we can take the square root of both sides to get x-2.5=±6.5. Solving for x, we get x=9 and x=-4, which are the same roots we found earlier using other techniques.

Completing the square is a powerful technique that can be used to solve not only quadratic equations but also other types of equations. It is a bit more involved than factoring, but it can be useful in cases where factoring is not possible.

Conclusion

In conclusion, we have explored three different techniques used to solve the equation x^2-36=5x. We found that the equation has two real roots, x=9 and x=-4, which can be obtained by factoring, using the quadratic formula, or completing the square. We also used the discriminant to determine the nature of the roots and graphed the equation to provide a visual representation. Depending on the situation, different techniques may be more appropriate, and it is always good to have multiple tools in our problem-solving toolbox.

Graphical Representation of the Equation

Graphical representation of an equation can provide a visual understanding of the equation. By plotting the graph of the equation, we can easily locate the roots of the equation and see the shape of the curve. The graph of the equation y=x^2-5x-36 is a parabola that opens upward. The x-coordinate of the vertex of the parabola can be found using the formula -b/2a, where a=1 and b=-5. Substituting these values, we get x=5/2. Therefore, the vertex of the parabola is at (5/2, -49/4).

The parabola intersects the x-axis at x=9 and x=-4, which are the roots of the equation. These points are also known as the x-intercepts of the parabola. The y-intercept of the parabola can be found by substituting x=0 into the equation. Substituting x=0, we get y=-36. Therefore, the y-intercept of the parabola is at (0, -36).

Conclusion

In conclusion, we have explored the different techniques used to solve the equation x^2-36=5x and have determined for what values of x the equation is true. We have found that the equation has two real roots, x=9 and x=-4. Graphically, the equation is represented by a parabola that opens upward and intersects the x-axis at x=9 and x=-4. The x-coordinate of the vertex of the parabola is 5/2, and the y-intercept is -36. The quadratic equation x^2-36=5x is a fundamental equation that has many practical applications in mathematics and physics. Mastering the techniques used to solve this equation is essential for students to excel in algebra and calculus.