Question

The value of \(\mathrm{k}\) for which the quadratic equation \(\mathrm{k} \mathrm{x}^{2}+1=\mathrm{k} \mathrm{x}+3 \mathrm{x}-11 \mathrm{x}^{2}\) has real and equal roots are (a) \(\\{-11,-3\\}\) (b) \(\\{5,7\\}\) (c) \(\\{5,-7\\}\) (d) \(\\{-5,-7\\}\)

Short answer

The values of k for which the quadratic equation has real and equal roots are \(k = \{5, -7\}\). So the correct answer is (c) \(\{5, -7\}\).

Step-by-step solution

Write the given quadratic equation in the standard form

The given quadratic equation is \(kx^2 + 1 = kx + 3x - 11x^2\). Let's rewrite this equation in the standard form: \(kx^2 + 11x^2 - kx - 3x + 1 = 0\). Combine the like terms: \((11 + k)x^2 - (k + 3)x + 1 = 0\).

Identify a, b, and c in the quadratic equation

Now that we have the quadratic equation in the standard form, we can identify the coefficients a, b, and c from the equation: \((11 + k)x^2 - (k + 3)x + 1 = 0\), Where, a = \(11 + k\), b = \(- (k + 3)\), and c = 1.

Find the discriminant (D) of the quadratic equation

For the quadratic equation to have real and equal roots, the discriminant (D) should be equal to zero. The discriminant is given by the formula: \(D = b^2 - 4ac\). Let's find the discriminant for the given quadratic equation using the coefficients a, b, and c identified above: \(D = (- (k + 3))^2 - 4(11 + k)(1)\). Now, we will set the discriminant equal to zero and solve for k: \((- (k + 3))^2 - 4(11 + k)(1) = 0\).

Solve for k

To find the values of k, solve the equation: \((- (k + 3))^2 - 4(11 + k)(1) = 0\). First, expand the equation: \((k^2 + 6k + 9) - 44 - 4k = 0\). Now, combine the like terms: \(k^2 + 2k - 35 = 0\). Next, factor the equation: \((k - 5)(k + 7) = 0\). Finally, solve for k: k = 5, -7

State the values of k for which the quadratic equation has real and equal roots

The values of k for which the quadratic equation has real and equal roots are: \(k = \{5, -7\}\). So the correct answer is (c) \(\{5, -7\}\).

Key concepts

Quadratic Equations

A quadratic equation is a polynomial equation of degree 2. This type of equation takes the general form, \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). The solutions to this equation are called the roots or zeros of the quadratic equation. Quadratic equations appear frequently in various fields, such as physics, engineering, and finance.

Solving a quadratic equation can be done using several methods, including:

• Factoring - if the equation can be factored into two binomials, it can be solved simply by setting each factor equal to zero.
• Quadratic Formula - a universal method that can find the roots of any quadratic equation: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Completing the square - involves reorganizing the equation into a perfect square trinomial.
• Graphical Method - plotting the equation on a graph and identifying the points where it crosses the x-axis.

Understanding the nature of the roots of a quadratic equation—whether they're real or imaginary, equal or distinct—depends significantly on the discriminant.

Discriminant

The discriminant is a crucial concept in understanding the nature of the roots of a quadratic equation. For any quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant \(D\) is defined by the formula:
\[D = b^2 - 4ac\]
The discriminant helps in determining how many and what kind of solutions a quadratic equation will have:

• Real and Equal Roots: If \(D = 0\), the quadratic equation has two real and equal roots. In this case, the graph of the quadratic equation touches the x-axis at exactly one point.
• Real and Unequal Roots: If \(D > 0\), the quadratic equation has two distinct real roots. The graph of the equation intersects the x-axis at two points.
• Complex Roots: If \(D < 0\), the equation has no real roots but two complex roots. The graph does not intersect the x-axis at all.

By calculating and analyzing the discriminant, we gain valuable insights into the behavior of the quadratic equation's graph and its intercepts with the x-axis.

Factorization

Factorization is one of the most straightforward techniques to solve a quadratic equation when the expression can be decomposed into simpler polynomials (usually binomials) multiplied together. In essence, you break down the quadratic equation, \(ax^2 + bx + c = 0\), into the form \((px + q)(rx + s) = 0\).

The key idea behind factorization is that if a product of terms equals zero, then at least one term must be zero. Therefore, the equation can be solved by setting each factor equal to zero, leading to:

• \(px + q = 0\) which gives \(x = -\frac{q}{p}\)
• \(rx + s = 0\) which gives \(x = -\frac{s}{r}\)

Factorization requires identifying two numbers that multiply to \(ac\) (the product of the coefficient of \(x^2\) and the constant term) and add up to \(b\) (the coefficient of \(x\)). In some instances, such as simple quadratics or those with integer roots, factorization offers a much quicker solution than using the quadratic formula.

Mastery of factorization can greatly simplify solving quadratic equations, especially before tackling more complex techniques.