Question

Find the real solutions of the equation \(2 x^{2}+7 x-4=0\).

Short answer

The solutions are \(x = \frac{1}{2}\) and \(x = -4\).

Step-by-step solution

- Identify coefficients

Recognize that the equation is a standard quadratic equation in the form of \(ax^2 + bx + c = 0\), where \(a = 2\), \(b = 7\), and \(c = -4\).

- Apply the quadratic formula

The quadratic formula is given by \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Substituting the values from the problem, we have \(a = 2\), \(b = 7\), and \(c = -4\).

- Compute the discriminant

Calculate the discriminant, which is \(b^2 - 4ac\). For this equation, it becomes: n\[7^2 - 4 \times 2 \times (-4) = 49 + 32 = 81\].

- Simplify the square root

Determine the square root of the discriminant. \( \sqrt{81} = 9\).

- Find the solutions

Substitute the values into the quadratic formula. \[x = \frac{-7 \pm 9}{4}\]. This results in two possible solutions: \[x_1 = \frac{-7 + 9}{4} = \frac{2}{4} = \frac{1}{2}\] and \[x_2 = \frac{-7 - 9}{4} = \frac{-16}{4} = -4\].

Key concepts

quadratic formula

The quadratic formula is a powerful tool for solving any quadratic equation of the form ewlineewline \text{quadratic:} \text{quadratic:} ewline\( ax^2 + bx + c = 0 \). It provides the solutions directly without needing to factor or complete the square. The formula is given by: ewline\[ x = \frac{-b \textbackslash pm \textbackslash sqrt{b^2 - 4ac}}{2a} \]. ewlineTo use the quadratic formula,

• Identify the coefficients
  • (a),
  • (b),
  • and (c)
• Substitute these values into the formula.
• Simplify the expression to find the values of x.

It involves computing the discriminant first (which we will cover in the next section), then finding the square root before performing the addition or subtraction. Don't forget to divide twice the coefficient of the quadratic term (2a). In the equation \textbackslash (2x^2 + 7x - 4 = 0), we have a = 2, b = 7, and c = -4. Plugging these values into the formula, we get \[ x = \frac{-7 \textbackslash pm \textbackslash sqrt{7^2 - 4 \textbackslash cdot 2 \textbackslash cdot (-4)}}{2 \textbackslash cdot 2} \]. Now, let's explore each step in detail.

discriminant

The discriminant is a vital part of the quadratic formula. It determines the nature of the solutions to the quadratic equation. The discriminant is the expression under the square root in the quadratic formula: \( b^2 - 4ac \). Here's why it's important:

• If the discriminant is positive (>0), there are two distinct real solutions.
• If the discriminant is zero (0), there is exactly one real solution (also called a repeated or double root).
• If the discriminant is negative (<0), there are no real solutions, but two complex solutions.

For our equation ewline \[ 2x^2 + 7x - 4 = 0 \], we compute the discriminant as follows: ewline\[ 7^2 - 4 \textbackslash cdot 2 \textbackslash cdot (-4) = 49 + 32 = 81 \]. Since our discriminant is 81, which is positive, this means our quadratic equation has two distinct real solutions.In the calculation step, it's important to be precise. Square the coefficient of x (\textbackslash b^2) and subtract four times the product of the coefficient of x^2 (\textbackslash 4a) and the constant term (\textbackslash c).Once you've calculated the discriminant, finding the solutions becomes straightforward.

solving quadratic equations

To solve quadratic equations using the quadratic formula, follow these steps:
\[ 1. Identify the coefficients: \] In our equation \( 2x^2 + 7x - 4 = 0 \), the coefficients are\( a = 2 \),\( b = 7 \),and \( c = -4 \).
\[ 2. Calculate the discriminant: \] Using the formula \( b^2 - 4ac \), we find \ \[ 7^2 - 4 \textbackslash cdot 2 \textbackslash cdot (-4) = 81 \]. This positive value tells us there are two real solutions.
\[ 3. Apply the quadratic formula: \] Substitute the values into \[ x = \frac{-b \textbackslash pm \textbackslash sqrt{b^2 - 4ac}}{2a} \], yielding: \[ x = \frac{-7 \textbackslash pm \textbackslash sqrt{81}}{4} \].
\[ 4. Simplify: \] Compute the square root of the discriminant, \ \( \textbackslash sqrt{81} = 9 \). Now substitute back into the formula,\[ -7 \textbackslash pm 9 \], and divide by \ \( 2 \textbackslash cdot a = 4 \). This results in: x=2/4 = 1/2 and x=-16/4 = -4Always remember to perform operations carefully and double-check your work. This method guarantees finding the correct solutions for any quadratic equation. If one follows these steps diligently, solving quadratic equations will be a straightforward task.