Problem 3 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If \(x\) be real, then least value of \(3 x^{2}+7 x+10\) is (a) 10 (b) \(10 / 3\) (c) \(7 / 3\) (d) \(71 / 12\)

Step-by-Step Solution

Answer

The least value is \(\frac{71}{12}\), matching option (d).

Step 1: Understand the problem

We need to find the least value (or minimum value) of the quadratic expression \(3x^2 + 7x + 10\) where \(x\) is a real number.

Step 2: Identify the formula for the minimum value of a quadratic equation

The general form of a quadratic equation is \(ax^2 + bx + c\). The minimum value of the quadratic equation occurs at \(x = -\frac{b}{2a}\).

Step 3: Apply formula to find vertex \(x\)

For our quadratic equation \(3x^2 + 7x + 10\), \(a = 3\) and \(b = 7\). Substituting into the formula gives the vertex at \(x = -\frac{7}{2 \times 3} = -\frac{7}{6}\).

Step 4: Calculate the minimum value using the vertex

Substitute \(x = -\frac{7}{6}\) back into the quadratic to find the minimum value: \(f\left(-\frac{7}{6}\right) = 3\left(-\frac{7}{6}\right)^2 + 7\left(-\frac{7}{6}\right) + 10\).

Step 5: Simplify the expression for the minimum value

Compute \(3\left(-\frac{7}{6}\right)^2 = 3 \times \frac{49}{36}\) and \(7 \times -\frac{7}{6} = -\frac{49}{6}\). This simplifies to \(\frac{49}{12} - \frac{49}{6} + 10\). Convert \(\frac{49}{6}\) to have a common denominator with \(\frac{49}{12}\) by getting \(\frac{98}{12}\).

Step 6: Final calculation and simplification

We now calculate: \(\frac{49}{12} - \frac{98}{12} + 10 = \frac{-49}{12} + 10 = \frac{-49}{12} + \frac{120}{12} = \frac{71}{12}\).

Step 7: Compare with given options

Our result \(\frac{71}{12}\) matches one of the given options (d) \(\frac{71}{12}\).

Key Concepts

Minimum Value

When dealing with quadratic equations, understanding the concept of minimum or maximum values is essential. A quadratic equation takes the general form \(ax^2 + bx + c\). For any quadratic equation opening upwards (where \(a > 0\)), there exists a minimum value of the quadratic expression. This value is found at the vertex of the parabola, which is the lowest point of the curve.To determine the position of this minimum value on the x-axis, we use the vertex formula \(x = -\frac{b}{2a}\). Once we find this specific \(x\) value, we substitute it back into the quadratic expression to compute the minimum value of the function.In our example \(3x^2 + 7x + 10\), after finding the vertex, substituting the \(x\) value of \(-\frac{7}{6}\) back into the equation, reveals the minimum value of the polynomial which is \(\frac{71}{12}\). This guides us toward understanding that the parabola reaches its least point of \(\frac{71}{12}\) when \(x\) is real.

Vertex Formula

The vertex formula is a powerful mathematical tool used to find the peak or lowest point of a parabola represented by a quadratic function \(ax^2 + bx + c\). When the parabola opens upwards (\(a > 0\)), the vertex represents the minimum point on the graph.The formula for finding this vertex is \(x = -\frac{b}{2a}\). This formula pinpoints the x-coordinate of the vertex, providing a precise location where the quadratic function achieves its minimum or maximum value.Using our given equation \(3x^2 + 7x + 10\), we applied \(a = 3\) and \(b = 7\) into the vertex formula. This gives us \(x = -\frac{7}{2 \times 3} = -\frac{7}{6}\) as the x-coordinate. Calculating this allows us to evaluate the function at \(-\frac{7}{6}\), confirming the minimum value of the quadratic expression.

Real Numbers

The realm of real numbers comprises all possible numbers along the number line, including both rational numbers (like fractions and integers) and irrational numbers (such as \(\sqrt{2}\) or \(\pi\)). Quadratic equations like the one explored, \(3x^2 + 7x + 10\), typically ensure that solutions are confined within the set of real numbers, especially when prompted to find the minimum or maximum values.Real numbers provide the necessary context for defining and solving these quadratic functions. Since real numbers ensure continuity, they allow for smooth curves like parabolas in the coordinate plane.In our specific problem, we were tasked to find the least value of the quadratic function for a real number \(x\). This requirement guaranteed that any manipulations, such as calculating the vertex or evaluating function values, adhere strictly to real number solutions, ensuring mathematical accuracy.