Solving for X-Intercepts in Quadratic Equations: y -3x^2 12x - 7

In this article, we will explore the process of finding the x-intercepts of a quadratic equation of the form y -3x^2 12x - 7. The x-intercepts, also known as the roots of the equation, are the points where the graph of the function intersects the x-axis, meaning the y-values are zero. This is a fundamental concept in algebra and has practical applications in many fields, including physics and engineering.

Understanding the Quadratic Equation

A quadratic equation is expressed in the form y ax^2 bx c, where a, b, and c are constants and a ≠ 0. In our case, a -3, b 12, and c -7.

Finding the X-Intercepts

To find the x-intercepts, we set y 0 and solve the equation for x. Starting with the given equation:

Step 1: Setting the Equation to Zero

0 -3x^2 12x - 7

Step 2: Simplifying the Equation

We can simplify the equation by multiplying through by -1 to make the leading coefficient positive:

Equation

3x^2 - 12x 7 0

Step 3: Using the Quadratic Formula

The quadratic formula is given by:

Formula

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

Where a 3, b -12, and c 7. Plugging these values into the formula, we get:

Step 4: Calculating the Discriminant

b^2 - 4ac (-12)^2 - 4 cdot 3 cdot 7 144 - 84 60

Step 5: Solving for X

x frac{-(-12) pm sqrt{60}}{2 cdot 3} frac{12 pm 2sqrt{15}}{6} frac{12}{6} pm frac{2sqrt{15}}{6} 2 pm frac{sqrt{15}}{3}

This gives us two solutions:

x 2 frac{sqrt{15}}{3} x 2 - frac{sqrt{15}}{3}

Conclusion

To summarize, the x-intercepts of the quadratic equation y -3x^2 12x - 7 are: x 2 frac{sqrt{15}}{3} x 2 - frac{sqrt{15}}{3}

Additional Resources

For a visual understanding, you can use Desmos graphing calculator. This tool is very useful for graphing quadratic and other polynomial functions, helping you visualize the intercepts and the overall behavior of the function.

Y-Intercept

The y-intercept of a quadratic equation occurs when x 0. Plugging x 0 into the original equation, we get:

Step 1: Substituting x 0

y -3(0)^2 12(0) - 7 -7

Therefore, the y-intercept is at the point (0, -7).

Frequently Asked Questions (FAQs)

How do you find the x-intercepts of a quadratic equation?
To find the x-intercepts, set y 0 and solve for x. Use the quadratic formula: x frac{-b pm sqrt{b^2 - 4ac}}{2a}. What is the y-intercept of the equation y -3x^2 12x - 7?
The y-intercept is the point where the graph crosses the y-axis. For the given equation, the y-intercept is at (0, -7). How do you determine the discriminant of a quadratic equation?
The discriminant is calculated as b^2 - 4ac. In the given equation, a -3, b 12, and c -7. The discriminant is 60, indicating two real and distinct roots.