Quadratic Equations
Discriminant and roots
Grade 11
Question:
<p>If <span>\(x, y\)</span> and <span>\(z\)</span> are real such that <span>\(x + y + z = 4\)</span>, <span>\(x^2 + y^2 + z^2 = 6\)</span>, find the range of <span>\(x\)</span>.</p>
<p>(a) <span>\((-1, 1)\)</span></p>
<p>(b) <span>\([0, 2]\)</span></p>
<p>(c) <span>\([\frac{2}{3}, 2]\)</span></p>
<p>(d) <span>\(\left[\frac{2}{\sqrt{3}}, \frac{2}{\sqrt{3}}\right]\)</span></p>
Step-by-Step Solution
Key Concept: Use the constraint equations to eliminate one variable and apply the discriminant condition for the resulting quadratic to ensure real solutions.
<p><strong>Step 1: Eliminate z</strong></p><p>From the constraints: <span>$x + y + (4 - x - y) = 4$</span> and <span>$x^2 + y^2 + (4 - x - y)^2 = 6$</span></p><p><strong>Step 2: Express as quadratic</strong></p><p><span>$y^2 + (x-4)y + x^2 - 4x + 5 = 0$</span></p><p><strong>Step 3: Apply discriminant condition</strong></p><p>For real <span>$y$</span>, we need <span>$D \geq 0$</span></p><p><span>$(x-4)^2 - 4(x^2 - 4x + 5) > 0$</span></p><p><span>$3x^2 - 8x + 4 \leq 0$</span></p><p><span>$(x - 2)(3x - 2) \leq 0$</span></p><p><span>$x \in \left[\frac{2}{3}, 2\right]$</span></p><p>∴ Answer is (c).</p>
Correct Answer: c