Question

Without solving, determine the character of the solutions of each equation in the complex number system. $$ 2 x^{2}+3 x=4 $$

Short answer

The equation has two distinct real solutions.

Step-by-step solution

Rewrite the equation

Rewrite the given equation in the standard quadratic form. The standard form is ax^{2} + bx + c = 0 . Here, move all terms to one side of the equation: 2x^{2} + 3x - 4 = 0 . identify a = 2 . ; b = 3 and c = -4

Calculate the Discriminant

The discriminant Δ is given by the formula Δ = b^{2} - 4ac . Substitute a = 2, b=3, and c = -4 into the formula: Δ = 3^{2} - 4(2)(-4) .

Simplify the Discriminant

Simplify the formula calculation Δ = 9 + 32 = 41 .

Determine the Character of the Solutions

Analyze the discriminant to determine the nature of the roots. Since Δ > 0 (41 > 0), the quadratic equation has two distinct real solutions.

Key concepts

Discriminant in Quadratic Equations

In quadratic equations, the discriminant is a key component that helps us understand the nature of the roots without actually solving the equation. The discriminant is part of the quadratic formula, which is: \ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \ Discriminant ( \Delta ) is represented by: \Delta = b^{2} - 4ac \ It is calculated from the coefficients of the quadratic equation in the form \( ax^{2} + bx + c = 0 \). The coefficients are: \ul * \(a\): Coefficient of \(x^{2}\) * \(b\): Coefficient of \(x\) * \(c\): Constant term Through analyzing the \(\Delta\), we can understand if the solutions are real or complex, and whether they are distinct or repeated.

Character of Solutions

The character of solutions in quadratic equations is determined by the value of the discriminant: \br \

• \(\Delta > 0\): This means there are two distinct real roots. This occurs because the square root of a positive number is real and non-zero.
• \(\Delta = 0\): This means there is exactly one real root, also known as a repeated or double root. The square root of zero is zero, which means the solutions coincide.
• \(\Delta < 0\): This means there are two complex roots. When the discriminant is negative, the square root of a negative number is imaginary, leading to complex solutions.
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For example, if we have an equation \(2x^{2} + 3x - 4 = 0\) and calculate its discriminant, we find \(\Delta = 41 \). Since \(41 > 0\), this tells us that the equation has two distinct real roots.

Roots of Quadratic Equations

The roots of a quadratic equation are the values of \(x\) that satisfy the equation. As mentioned, the roots can be found using the quadratic formula: \br \ x = \frac{-b \pm \sqrt{\Delta}}{2a} Where \Delta = b^{2} - 4ac. \ Depending on the sign of \Delta\, the roots fall into different categories: \br \

• **Real and distinct roots**: Occur if \(\Delta > 0\). For instance, an equation like \(2x^{2} + 3x - 4 = 0\) with a discriminant of 41 has two different real roots.
• **Real and repeated roots**: Occur if \(\Delta = 0\). The equation \(x^{2} - 2x + 1 = 0\) has \(\Delta = 0\), resulting in one real root (\(x = 1\)).
• **Complex roots**: Occur if \(\Delta < 0\). The equation \(x^{2} + x + 1 = 0\) has \(\Delta = -3\), leading to roots in the form of complex numbers (\(x = \frac{-1 \pm i\sqrt{3}}{2}\)).
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Remembering the role of the discriminant is crucial in predicting the nature of quadratic solutions even before solving the equation.