What is the Quadratic Formula? – Find the Vertex of y = 4×2-5x-12 – All You Need to Know

The equation 4x ^ 2 – 5x – 12 = 0 cannot be easily factored since the middle term doesn’t fit the other two terms. However, it can be solved using the quadratic formula: In this case, =4,a=4, =−5,b= −5, and = −12.c=−12. Plugging these values into the formula, we find: Look over the complete guide and make your mind swift;

Find the Vertex of y = 4×2-5x-12

Even though we do not plot “y” until after plotting the coefficient of the first term, 4, we already know this is true (greater than 0). Several parabolas have a vertical line of symmetry that passes through the vertex of each parabola.

As a result of this symmetry, the line of symmetry can move through the midpoint of the two x-intercepts (roots) of the parabola, which will, for example, be the midpoint of the two x-intercepts. In other words, if there are two real solutions to the parabola, the equation is valid. Various real-life situations can be modeled using parabolas, such as the height above the ground of an object thrown upwards after some time.

Several things can be learned about the parabola’s vertex. For example, we can learn about the maximum height that an object can achieve when thrown upwards.

Also, the x-coordinate of the vertex of a parabola, Ax2+Bx+C, can be found by multiplying -B/(2A) by the x-coordinate of the vertex. 0.6250 is the coordinate of the x-axis in our case. By plugging into the parabola formula 0.6250 for x, we can calculate the y -y-coordinate using the following formula: 

y = 4.0 0.62 0.62 – 5.0 * 0.62 – 12.0

or  y = -13.562 – explain it in detail

What is the Quadratic Formula?

The quadratic formula is a fundamental tool in algebra used to find the solutions to a quadratic equation. It is a formula derived from completing the square technique and provides the roots or zeros of the quadratic equation. The formula is of the form:

(x – sqrt((b2 – 4ac) / (4a))) / 2a

Where:

• a, b, and c are known real or complex numbers with ≠ 0, and a <0xEE><0x80><0xA0> = 0
• x represents an unknown variable
• The formula states that the solutions for x are given by

The quadratic formula is particularly useful in solving quadratic equations that cannot be easily factored. By substituting the coefficients a, b, and c into the formula, one can calculate the values of x that satisfy the equation.

The quadratic formula is a powerful mathematical tool that simplifies finding solutions to quadratic equations. It eliminates the need for factoring and allows for quick and efficient calculations. The formula can be applied in various mathematical contexts, including finding the maximum or minimum values of a function, analyzing geometric shapes, and solving systems of equations.

In summary, the quadratic formula is an essential algebra tool used to find solutions to quadratic equations. It simplifies the process of solving quadratic equations that cannot be easily factored and is an essential mathematical tool for solving a wide range of problems.

Determining the Number of Solutions of a Quadratic Equation

To analyze and determine the number of solutions of a quadratic equation using the discriminant, it is crucial to follow specific rules based on the value of the discriminant. These rules provide valuable insights into the nature and number of solutions that the equation possesses. When evaluating a quadratic equation, the discriminant is calculated using the formula:

Discriminant = (b2 – 4ac) / (4a)

Where a and b are the coefficients of the quadratic term and the constant term, respectively, and c is the value of the quadratic term.

Based on the evaluation of the discriminant, the following rules are applicable:

• Positive Discriminant: If the discriminant is positive, two distinct real solutions exist to the quadratic equation. These solutions can live on the actual number line and as complex numbers.
• Zero Discriminant: If the discriminant is zero, there is one real solution (a repeated root). This occurs when the equation possesses a perfect square trinomial factor.
• Negative Discriminant: If the discriminant is negative, there are two complex conjugate solutions to the quadratic equation. These solutions can be expressed in terms of the natural and imaginary parts of the complex number. By calculating the discriminant and analyzing its value, it becomes straightforward to determine the nature and number of solutions that a quadratic equation possesses.

What is the Discriminant in the Quadratic Formula

The discriminant in the quadratic formula is ac, where a, b, and c represent the coefficients of the quadratic equation ax^2+bx+c. The discriminant measures the “size” of the quadratic equation and plays a crucial role in determining the nature and number of solutions to the equation.

If the discriminant is positive, the equation has two distinct real solutions. This means that the solutions can be found by dividing the square root of the sum of the squares of the coefficients by the square of the sum or difference of the squares of the coefficients. If the discriminant is zero, the equation has one real solution. In this case, the solutions are the numbers that solve the equation when set equal to zero. If the discriminant is negative, the equation has two complex conjugate solutions.

Complex conjugate solutions refer to solutions that are related to an imaginary number. They are solutions of the equation that are a pair of complex numbers where one of them is multiplied by the imaginary unit j. In quadratic equations, complex conjugate solutions occur whenever the equation has a root on the imaginary axis.

A discriminant determines the type and number of solutions to a quadratic equation by determining the nature of the discriminant in the quadratic formula. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex conjugate solutions.

How to Use the Discriminant to Determine the Nature of the Roots of a Quadratic Equation

The discriminant of a quadratic equation is a mathematical expression that helps us determine the nature and number of the roots of a quadratic equation. By analyzing the discriminant, we can determine whether the roots are real, imaginary, or complex numbers and the number of real roots.

The formula for the Discriminant

The following formula gives the discriminant of a quadratic equation:

Discriminant = b^2 – 4ac;

Where:

• a and c are the coefficients of the equation’s leading term and the constant term, respectively.
• b is the value of the leading coefficient.

Real Roots

If the discriminant is positive, the quadratic equation has two real roots. If the value of the discriminant is equal to zero, the quadratic equation has two distinct real roots; if it is negative, the quadratic equation has one real root.

Imaginary Roots

The quadratic equation has two distinct imaginary roots if the discriminant is negative. If the value of the discriminant is equal to zero, the quadratic equation has one real root and one imaginary root.

Complex Roots

If the discriminant is non-negative, the quadratic equation may have complex roots. To determine the number of complex roots, we can use the following formula:

Number of Complex Roots = 2(Discriminant – 4)

If the discriminant is positive, there are zero complex roots. If the value of the discriminant is less than or equal to 4, the quadratic equation has one complex root. If the discriminant is greater than 4, the quadratic equation has two or more complex roots.

Consider a quadratic equation:

x^2 – 4x + 2 = 0

To find the discriminant, we substitute the values:

Discriminant = b^2 – 4ac

Discriminant = (2)^2 – 4(4)(2) + 2

Discriminant = 16 – 16 + 2

Discriminant = 2

The discriminant in this case is number 2. It is evident that the quadratic equation has two real roots since the discriminant is positive. As a result of solving the equation, the roots are found as follows:

x^2 – 4x + 2 = 0

(x – 2)(x – 1) = 0

x = 2 or x = 1

Therefore, the roots of the quadratic equation are 2 and 1. When determining the nature and number of the roots of a quadratic equation, the discriminant is one of the most valuable tools. It is possible to determine whether the roots of the discriminant are real, imaginary, or complex numbers by analyzing the discriminant and the number of real roots that exist. A systematic method for solving quadratic equations can be developed by using the formula for the discriminant and the rules for determining the roots’ nature.

Do read more topics on education.