Question

If the ratio of the roots of the quadratic equation \(2 x^{2}+16 x+3 k=0\) is \(4: 5\) then \(k=\) (a) \([(2560) /(243)]\) (b) \(\overline{[(243) /(2560)]}\) (c) \([(-2560) /(243)]\) (d) \([(-243) /(2560)]\)

Short answer

The correct answer is \(k= \frac{(-2560)}{243}\).

Step-by-step solution

Identify the sum and product of the roots

Given the quadratic equation \(2x^2 + 16x + 3k = 0\), let the roots be \(α\) and \(β\). According to the problem, the ratio of the roots is given by \(α:β = 4:5\). Thus, we can express the roots as \(α = 4m\) and \(β = 5m\), where \(m\) is a constant.

Apply Vieta's Formulas

Using Vieta's formulas, the sum and product of the roots can be expressed in terms of the coefficients of the quadratic equation as follows: Sum of the roots: \(α + β = \frac{-B}{A}\) Product of the roots: \(α × β = \frac{C}{A}\) Where A, B, and C are the coefficients of the quadratic equation \(2x^2 + 16x + 3k = 0\).

Calculate the sum and product of the roots

Using the expressions for the roots \(α = 4m\) and \(β = 5m\), we can calculate the sum and product of the roots: Sum of the roots: \(α + β = 4m + 5m = 9m\) Product of the roots: \(α × β = 4m × 5m = 20m^2\)

Set up and solve equations using Vieta's Formulas

Now, we will use Vieta's formulas to set up equations: Sum of the roots: \(\frac{-B}{A} = 9m\) ⟹ \(\frac{-16}{2} = 9m\) Product of the roots: \(\frac{C}{A} = 20m^2\) ⟹ \(\frac{3k}{2} = 20m^2\) Solving the first equation for \(m\): \(-8 = 9m\) \(m = -\frac{8}{9}\) Now, substitute the value of \(m\) in the second equation: \(\frac{3k}{2} = 20\left( -\frac{8}{9} \right)^2\)

Solve for k and find the correct answer

Now, we will solve the equation for the value of k: \(3k = 2 × 20\left( -\frac{8}{9} \right)^2\) \(k = \frac{2}{3} × 20\left( -\frac{8}{9} \right)^2\) \(k = \frac{(-2560)}{243}\) Comparing the calculated value of k with the given options, we can see that our answer matches option (c). Therefore, the correct answer is \(k= \boxed{\frac{(-2560)}{243}}\).

Key concepts

Vieta's Formulas

Understanding Vieta's formulas is key to solving many problems involving quadratic equations. These formulas allow us to use the coefficients of the equation to find the sum and product of its roots. For any quadratic equation in the standard form \(ax^2 + bx + c = 0\), Vieta's formulas state that:

• The sum of the roots \(\alpha + \beta\) is \(-\frac{b}{a}\).
• The product of the roots \(\alpha \times \beta\) is \(\frac{c}{a}\).

Knowing these relationships makes it possible to find or confirm the roots of the quadratic without solving it directly.

Sum and Product of Roots

In order to use Vieta's formulas, we must first understand how they relate to the sum and product of the roots of the quadratic equation. From the step-by-step solution, for the quadratic equation \(2x^2 + 16x + 3k = 0\), the coefficients are \(A = 2\), \(B = 16\), and \(C = 3k\).

• To find the sum of the roots, we apply the formula \(\alpha + \beta = -\frac{B}{A} = -\frac{16}{2}\), which equals \(-8\).
• For the product of the roots, use \(\alpha \times \beta = \frac{C}{A} = \frac{3k}{2}\).

Inserting these values allows you to derive equations for the roots' sum and product, which can be used along with the given ratio to determine the unknown \(k\).

Ratio of Roots

The problem introduces a ratio of roots, \(\alpha : \beta = 4 : 5\), which implies that \(\alpha = 4m\) and \(\beta = 5m\) for some constant \(m\). Using the sum of the roots formula, we find that \(9m = -8\), leading to \(m = -\frac{8}{9}\). This provides enough information to solve for \(m\), which is crucial because it ties directly to the product of the roots.

• With \(\alpha \times \beta = 20m^2\) and \(\frac{3k}{2} = 20m^2\), substituting \(m\) gives the equation \(\frac{3k}{2} = 20 \left(-\frac{8}{9}\right)^2\).
• Solving for \(k\) results in \(k = \frac{(-2560)}{243}\).

Thus, the ratio of the roots, combined with Vieta's formulas, enables you to determine the constant \(k\) fully.