Question

1. accurate to the nearest hundredth, the larger root of $2x^2 + 5x - 1 = 0$ is

(1) 1.85
(3) 2.47
(2) 0.19
(4) -3.56

Explanation:

Step1: Recall quadratic formula

For a quadratic equation \(ax^2 + bx + c = 0\), the roots are given by \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\).
Here, \(a = 2\), \(b = 5\), \(c=-1\).

Step2: Calculate discriminant

Discriminant \(D=b^2 - 4ac=(5)^2-4\times2\times(-1)=25 + 8 = 33\).

Step3: Find the roots

First root: \(x_1=\frac{-5+\sqrt{33}}{2\times2}=\frac{-5+\sqrt{33}}{4}\)
Second root: \(x_2=\frac{-5-\sqrt{33}}{4}\)
Since \(\sqrt{33}\approx5.7446\),
\(x_1=\frac{-5 + 5.7446}{4}=\frac{0.7446}{4}\approx0.18615\approx0.19\) (rounded to nearest hundredth)
Wait, no, wait. Wait, maybe I made a mistake. Wait, the quadratic is \(2x^2+5x - 1=0\), so \(a = 2\), \(b = 5\), \(c=-1\). So the roots are \(\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{-5\pm\sqrt{25 + 8}}{4}=\frac{-5\pm\sqrt{33}}{4}\). Now, \(\sqrt{33}\approx5.7446\). So \(\frac{-5 + 5.7446}{4}=\frac{0.7446}{4}\approx0.186\approx0.19\) and \(\frac{-5-5.7446}{4}=\frac{-10.7446}{4}\approx - 2.686\). Wait, but the options have 0.19 (option 2), 1.85, 2.47, - 3.56. Wait, maybe I messed up the quadratic formula. Wait, no, the quadratic formula is correct. Wait, maybe the equation is \(2x^2-5x - 1=0\)? Wait, no, the problem says \(2x^2 + 5x - 1=0\). Wait, let me check the options again. The options are (1) 1.85, (2) 0.19, (3) 2.47, (4) - 3.56. Wait, maybe I made a mistake in calculation. Wait, let's recalculate. \(\sqrt{33}\approx5.7446\). So \(\frac{-5+\sqrt{33}}{4}=\frac{-5 + 5.7446}{4}=\frac{0.7446}{4}\approx0.186\approx0.19\), and \(\frac{-5-\sqrt{33}}{4}=\frac{-5 - 5.7446}{4}=\frac{-10.7446}{4}\approx - 2.686\). But - 2.686 is not in the options. Wait, maybe the quadratic is \(2x^2-5x - 1=0\)? Let's try that. If \(a = 2\), \(b=-5\), \(c=-1\). Then discriminant \(D = 25+8 = 33\), roots \(\frac{5\pm\sqrt{33}}{4}\). Then \(\frac{5 + 5.7446}{4}=\frac{10.7446}{4}\approx2.686\), \(\frac{5 - 5.7446}{4}=\frac{-0.7446}{4}\approx - 0.186\). Still not matching. Wait, maybe the equation is \(x^2+5x - 2=0\)? No, the problem says \(2x^2+5x - 1=0\). Wait, maybe the options are wrong, or I made a mistake. Wait, let's check the calculation again. \(\sqrt{33}\approx5.7446\). So \(\frac{-5 + 5.7446}{4}=\frac{0.7446}{4}=0.18615\approx0.19\) (option 2), and \(\frac{-5 - 5.7446}{4}=\frac{-10.7446}{4}=-2.686\). But the options have - 3.56. Wait, maybe the quadratic is \(2x^2+5x + 1=0\)? No, the problem says \(2x^2 + 5x - 1=0\). Wait, maybe I miscalculated the discriminant. \(b^2-4ac=25-4\times2\times(-1)=25 + 8 = 33\), that's correct. \(\sqrt{33}\approx5.7446\), correct. Then \(\frac{-5 + 5.7446}{4}\approx0.186\approx0.19\), and \(\frac{-5 - 5.7446}{4}\approx - 2.686\). But the options have - 3.56. Wait, maybe the equation is \(2x^2+5x - 3=0\)? Let's see, if it's \(2x^2+5x - 3=0\), then \(a = 2\), \(b = 5\), \(c=-3\), discriminant \(25+24 = 49\), roots \(\frac{-5\pm7}{4}\), so \(\frac{2}{4}=0.5\) and \(\frac{-12}{4}=-3\), not matching. Wait, the options are (1) 1.85, (2) 0.19, (3) 2.47, (4) - 3.56. Let's check option (2) 0.19, which is close to our calculation of 0.186. So maybe the answer is (2) 0.19. Wait, but let's check again. Wait, maybe I made a mistake in the sign of \(c\). If the equation was \(2x^2-5x - 1=0\), then \(a = 2\), \(b=-5\), \(c=-1\), discriminant \(25 + 8 = 33\), roots \(\frac{5\pm\sqrt{33}}{4}\), \(\frac{5 + 5.7446}{4}\approx2.686\), \(\frac{5 - 5.7446}{4}\approx - 0.186\). Still not matching. Wait, the options have 0.19, which is close to our first root. So maybe the answer is (2) 0.19.

Answer:

(2) 0.19